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- The document discusses linear time-invariant (LTI) systems, which are systems that are both linear and time-invariant. - The behavior of an LTI system is completely defined by its impulse response. The impulse response is the output of the system when given an impulse as input. - Convolution is used to find the output of an LTI system for any input. The output is the summation of each input multiplied by the impulse response shifted in time. Convolution allows predicting the long-term behavior of an LTI system.

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19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Signals and Systems UNIT 2 Ripal Patel Assistant Professor, Dr.Ambedkar Institute of Technology, Bangalore. ripal.patel@dr-ait.org December 1, 2020
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs.
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. • Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically.
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. • Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically. • Used to predict long-term behavior in a system
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. • Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically. • Used to predict long-term behavior in a system • The behavior of an LTI system is completely defined by its impulse response
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Function The discrete version of impulse function is defined by δ(n) = 1, n = 0 0, n 6= 0 The continuous time version of impulse function, δ(t) = 1, t = 0 0, t 6= 0
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Response • The impulse response” of a system, h[n], is the output that it produces in response to an impulse input. Definition: if and only if x[n] = δ[n] then y[n] = h[n]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Response • The impulse response” of a system, h[n], is the output that it produces in response to an impulse input. Definition: if and only if x[n] = δ[n] then y[n] = h[n] • Given the system equation, the impulse response can be found out just by feeding x[n] = δ[n] into the system.
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Response Example • Consider the system y[n] = 1 2 (x[n] + x[n − 1]) • Suppose we insert an impulse: x[n] = δ[n] • Then whatever we get at the output, by Definition, is the impulse response. In this case it is h[n] = 1 2 (δ[n] + δ[n − 1]) = 0.5, n = 0, 1 0, otherwise
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • where, h[n]=impulse response of LTI system x[n]=Input Signal
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • where, h[n]=impulse response of LTI system x[n]=Input Signal • y[n] = ∞ X k=−∞ x[k]h[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • where, h[n]=impulse response of LTI system x[n]=Input Signal • y[n] = ∞ X k=−∞ x[k]h[n − k] • input-excitation output-response
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Example Find the response y[n] of following LTI system.
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • • •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • • • •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • • • • • y(n) = [...0, 1, 4 ↑ , 9, 11, 8, 2, 0, ...]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4 • For n=-1 y[−1] = 3 X k=0 x[k]h[−1 − k] = x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4 • For n=-1 y[−1] = 3 X k=0 x[k]h[−1 − k] = x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1 • For n=0 y[0] = 3 X k=0 x[k]h[0 − k] = x[0]h[0]+x[1]h[−1]+x[2]h[−2] = (1x2)+(2x1)+(3x0) = 4
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4 • For n=-1 y[−1] = 3 X k=0 x[k]h[−1 − k] = x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1 • For n=0 y[0] = 3 X k=0 x[k]h[0 − k] = x[0]h[0]+x[1]h[−1]+x[2]h[−2] = (1x2)+(2x1)+(3x0) = 4 • Likewise for all the values of n y(n) = [...0, 1, 4 ↑ , 9, 11, 8, 2, 0, ...]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) x1(n) = [1 ↑ , 2, 3] x2(n) = [2 ↑ , 1, 4] y[n] = ∞ X k=−∞ x1[k]x2[n − k] • Size of x1(n)=A=3, Size of x2(n)=B=3 Length of y(n)=A+B-1=3+3-1=5
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) x1(n) = [1 ↑ , 2, 3] x2(n) = [2 ↑ , 1, 4] y[n] = ∞ X k=−∞ x1[k]x2[n − k] • Size of x1(n)=A=3, Size of x2(n)=B=3 Length of y(n)=A+B-1=3+3-1=5 • x1(n) is starting from 0 index n1=0 x2(n) is starting from 0 index n2=0 n1 + n2 = 0, range of n=0 to 4
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • For n=0 y[0] = 2 X k=0 x1[k]x2[−k] = x[0]x2[0] + x1[1]x2[−1] + x1[2]x2[−2] = (1x2) + (2x0) + (3x0) = 2
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • For n=0 y[0] = 2 X k=0 x1[k]x2[−k] = x[0]x2[0] + x1[1]x2[−1] + x1[2]x2[−2] = (1x2) + (2x0) + (3x0) = 2 • Likewise for all the values of n y(n) = [2 ↑ , 5, 12, 11, 12]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k] • = P∞ k=−∞ αku[k]βnβ−ku[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k] • = P∞ k=−∞ αku[k]βnβ−ku[n − k] • = βn P∞ k=−∞ αku[k]β−ku[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k] • = P∞ k=−∞ αku[k]βnβ−ku[n − k] • = βn P∞ k=−∞ αku[k]β−ku[n − k] • = βn P∞ k=−∞ (α β )k u[k]u[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • u[k]u[n − k] = 1, 0 ≤ k ≤ n, n ≥ 0 •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • = βn Pn k=0 (α β )k
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • = βn Pn k=0 (α β )k • = βn [ (α β )n+1 − 1 (α β ) − 1 ]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • = βn Pn k=0 (α β )k • = βn [ (α β )n+1 − 1 (α β ) − 1 ] • 1 β − α [βn+1 − αn+1 ]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k] • = P∞ k=−∞(3 4)ku[k]u[n − k]
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k] • = P∞ k=−∞(3 4)ku[k]u[n − k] • = Pn k=0 (3 4) k
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k] • = P∞ k=−∞(3 4)ku[k]u[n − k] • = Pn k=0 (3 4) k • = ( 3 4 )n+1−1 3 4 −1
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • Convolution Integral between two continuous signals x(t) and h(t) y(t) = x(t) ∗ h(t) = Z ∞ −∞ x(τ)h(t − τ)dτ
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • Convolution Integral between two continuous signals x(t) and h(t) y(t) = x(t) ∗ h(t) = Z ∞ −∞ x(τ)h(t − τ)dτ • y(t)) = x(t) ∗ h(t) = h(t) ∗ x(t)
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Graphical Methods) Suppose the input x(t) and impulse response h(t) of a LTI system are given by x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3)
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral h(t)
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral Convolution :
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ • Case 1: t + 2 0, do not overlap hence zero.
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ • Case 1: t + 2 0, do not overlap hence zero. • Case 2 = R t+2 0 e−2τ x1dτ
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ • Case 1: t + 2 0, do not overlap hence zero. • Case 2 = R t+2 0 e−2τ x1dτ • = [e−2τ −2 ]t+2 0 = 1 2 − 1 2e−2(t+2)
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3) • x(t) = 1, t = 1, 2, 3
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3) • x(t) = 1, t = 1, 2, 3 • h(t) =    1 for 1 ≤ t ≤ 3 −1 for −1 ≤ t 1 0 otherwise
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3) • x(t) = 1, t = 1, 2, 3 • h(t) =    1 for 1 ≤ t ≤ 3 −1 for −1 ≤ t 1 0 otherwise • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • •
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response for Continuous time system
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response Example
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response Example
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Differential and Difference equation representation of LTI systems
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Differential and Difference equation representation of LTI systems
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Recursive evaluation of difference equation
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Solving Differential and Difference equations
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution for CT system
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution for DT system
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution Example
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution Example
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution Example
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems The Particular Solution
19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems The End

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Signals_Systems_Unit2_part2.pdf

  • 1. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Signals and Systems UNIT 2 Ripal Patel Assistant Professor, Dr.Ambedkar Institute of Technology, Bangalore. ripal.patel@dr-ait.org December 1, 2020
  • 2. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant
  • 3. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs.
  • 4. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. • Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically.
  • 5. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. • Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically. • Used to predict long-term behavior in a system
  • 6. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems LTI systems • A class of systems used in signals and systems that are both linear and time-invariant • Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. • Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically. • Used to predict long-term behavior in a system • The behavior of an LTI system is completely defined by its impulse response
  • 7. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Function The discrete version of impulse function is defined by δ(n) = 1, n = 0 0, n 6= 0 The continuous time version of impulse function, δ(t) = 1, t = 0 0, t 6= 0
  • 8. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Response • The impulse response” of a system, h[n], is the output that it produces in response to an impulse input. Definition: if and only if x[n] = δ[n] then y[n] = h[n]
  • 9. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Response • The impulse response” of a system, h[n], is the output that it produces in response to an impulse input. Definition: if and only if x[n] = δ[n] then y[n] = h[n] • Given the system equation, the impulse response can be found out just by feeding x[n] = δ[n] into the system.
  • 10. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Impulse Response Example • Consider the system y[n] = 1 2 (x[n] + x[n − 1]) • Suppose we insert an impulse: x[n] = δ[n] • Then whatever we get at the output, by Definition, is the impulse response. In this case it is h[n] = 1 2 (δ[n] + δ[n − 1]) = 0.5, n = 0, 1 0, otherwise
  • 11. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • where, h[n]=impulse response of LTI system x[n]=Input Signal
  • 12. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • where, h[n]=impulse response of LTI system x[n]=Input Signal • y[n] = ∞ X k=−∞ x[k]h[n − k]
  • 13. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • where, h[n]=impulse response of LTI system x[n]=Input Signal • y[n] = ∞ X k=−∞ x[k]h[n − k] • input-excitation output-response
  • 14. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Example Find the response y[n] of following LTI system.
  • 15. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) •
  • 16. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • •
  • 17. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • • •
  • 18. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • • • •
  • 19. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Graphical method) • • • • • y(n) = [...0, 1, 4 ↑ , 9, 11, 8, 2, 0, ...]
  • 20. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6
  • 21. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4
  • 22. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4 • For n=-1 y[−1] = 3 X k=0 x[k]h[−1 − k] = x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1
  • 23. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4 • For n=-1 y[−1] = 3 X k=0 x[k]h[−1 − k] = x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1 • For n=0 y[0] = 3 X k=0 x[k]h[0 − k] = x[0]h[0]+x[1]h[−1]+x[2]h[−2] = (1x2)+(2x1)+(3x0) = 4
  • 24. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • Size of x(n)=A=4, Size of h(n)=B=3 Length of y(n)=A+B-1=4+3-1=6 • x(n) is starting from 0 index n1=0 h(n) is starting from -1 index n2=-1 n1 + n2 = −1, range of n=-1 to 4 • For n=-1 y[−1] = 3 X k=0 x[k]h[−1 − k] = x[0]h[−1] + x[1]h[−2] = (1x1) + (2x0) = 1 • For n=0 y[0] = 3 X k=0 x[k]h[0 − k] = x[0]h[0]+x[1]h[−1]+x[2]h[−2] = (1x2)+(2x1)+(3x0) = 4 • Likewise for all the values of n y(n) = [...0, 1, 4 ↑ , 9, 11, 8, 2, 0, ...]
  • 25. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) x1(n) = [1 ↑ , 2, 3] x2(n) = [2 ↑ , 1, 4] y[n] = ∞ X k=−∞ x1[k]x2[n − k] • Size of x1(n)=A=3, Size of x2(n)=B=3 Length of y(n)=A+B-1=3+3-1=5
  • 26. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) x1(n) = [1 ↑ , 2, 3] x2(n) = [2 ↑ , 1, 4] y[n] = ∞ X k=−∞ x1[k]x2[n − k] • Size of x1(n)=A=3, Size of x2(n)=B=3 Length of y(n)=A+B-1=3+3-1=5 • x1(n) is starting from 0 index n1=0 x2(n) is starting from 0 index n2=0 n1 + n2 = 0, range of n=0 to 4
  • 27. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • For n=0 y[0] = 2 X k=0 x1[k]x2[−k] = x[0]x2[0] + x1[1]x2[−1] + x1[2]x2[−2] = (1x2) + (2x0) + (3x0) = 2
  • 28. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum (Analytical method) • For n=0 y[0] = 2 X k=0 x1[k]x2[−k] = x[0]x2[0] + x1[1]x2[−1] + x1[2]x2[−2] = (1x2) + (2x0) + (3x0) = 2 • Likewise for all the values of n y(n) = [2 ↑ , 5, 12, 11, 12]
  • 29. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n]
  • 30. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k]
  • 31. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k]
  • 32. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k] • = P∞ k=−∞ αku[k]βnβ−ku[n − k]
  • 33. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k] • = P∞ k=−∞ αku[k]βnβ−ku[n − k] • = βn P∞ k=−∞ αku[k]β−ku[n − k]
  • 34. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • y[n] = x1[n] ∗ x2[n] • = P∞ k=−∞ x1[k]x2[n − k] • = P∞ k=−∞ αku[k]βn−ku[n − k] • = P∞ k=−∞ αku[k]βnβ−ku[n − k] • = βn P∞ k=−∞ αku[k]β−ku[n − k] • = βn P∞ k=−∞ (α β )k u[k]u[n − k]
  • 35. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum •
  • 36. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum • u[k]u[n − k] = 1, 0 ≤ k ≤ n, n ≥ 0 •
  • 37. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • = βn Pn k=0 (α β )k
  • 38. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • = βn Pn k=0 (α β )k • = βn [ (α β )n+1 − 1 (α β ) − 1 ]
  • 39. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum Convolute the given sequences x1[n] = αnu[n] and x2[n] = βnu[n] • = βn Pn k=0 (α β )k • = βn [ (α β )n+1 − 1 (α β ) − 1 ] • 1 β − α [βn+1 − αn+1 ]
  • 40. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
  • 41. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k]
  • 42. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k] • = P∞ k=−∞(3 4)ku[k]u[n − k]
  • 43. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k] • = P∞ k=−∞(3 4)ku[k]u[n − k] • = Pn k=0 (3 4) k
  • 44. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Sum The system is characterized by an impulse response h[n] = ( 3 4 )n u[n] Find the step response of the system. Also evaluate the output of the system at n = ±5 • y[n] = x[n] ∗ h[n] = h[n] ∗ x[n] • = P∞ k=−∞ h[k]x[n − k] • = P∞ k=−∞(3 4)ku[k]u[n − k] • = Pn k=0 (3 4) k • = ( 3 4 )n+1−1 3 4 −1
  • 45. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • Convolution Integral between two continuous signals x(t) and h(t) y(t) = x(t) ∗ h(t) = Z ∞ −∞ x(τ)h(t − τ)dτ
  • 46. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • Convolution Integral between two continuous signals x(t) and h(t) y(t) = x(t) ∗ h(t) = Z ∞ −∞ x(τ)h(t − τ)dτ • y(t)) = x(t) ∗ h(t) = h(t) ∗ x(t)
  • 47. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Graphical Methods) Suppose the input x(t) and impulse response h(t) of a LTI system are given by x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3)
  • 48. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral h(t)
  • 49. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral Convolution :
  • 50. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ
  • 51. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ
  • 52. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ • Case 1: t + 2 0, do not overlap hence zero.
  • 53. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ • Case 1: t + 2 0, do not overlap hence zero. • Case 2 = R t+2 0 e−2τ x1dτ
  • 54. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = e−2t u(t) h(t) = u(t + 2) • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ • R ∞ −∞ e−2τ u(τ)u(t − τ + 2)dτ • Case 1: t + 2 0, do not overlap hence zero. • Case 2 = R t+2 0 e−2τ x1dτ • = [e−2τ −2 ]t+2 0 = 1 2 − 1 2e−2(t+2)
  • 55. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3) • x(t) = 1, t = 1, 2, 3
  • 56. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3) • x(t) = 1, t = 1, 2, 3 • h(t) =    1 for 1 ≤ t ≤ 3 −1 for −1 ≤ t 1 0 otherwise
  • 57. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral (Analytical Methods) x(t) = 2u(t − 1) − 2u(t − 3) h(t) = u(t + 1) − 2u(t − 1) + u(t − 3) • x(t) = 1, t = 1, 2, 3 • h(t) =    1 for 1 ≤ t ≤ 3 −1 for −1 ≤ t 1 0 otherwise • y(t) = x(t) ∗ h(t) = R ∞ −∞ x(τ)h(t − τ)dτ
  • 58. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral •
  • 59. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • •
  • 60. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral • •
  • 61. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Convolution Integral
  • 62. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response
  • 63. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response for Continuous time system
  • 64. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response Example
  • 65. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Step Response Example
  • 66. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Differential and Difference equation representation of LTI systems
  • 67. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Differential and Difference equation representation of LTI systems
  • 68. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Recursive evaluation of difference equation
  • 69. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Solving Differential and Difference equations
  • 70. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution for CT system
  • 71. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution for DT system
  • 72. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution Example
  • 73. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution Example
  • 74. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems Homogeneous Solution Example
  • 75. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems The Particular Solution
  • 76. 19EC34 Ripal Patel Time domain representation of LTI System Linear time-invariant systems (LTI systems) Impulse Response Convolution Sum Convolution Sum (Finite Sequences) Convolution Sum (Infinite Sequences) Convolution Integral Convolution Integral (Finite signals) Step Response Differential and Difference equation representation of LTI systems The End