Dsp class 3
Download as PPTX, PDF0 likes178 views
This document provides an introduction to discrete time signals and systems. It defines discrete time signals as functions of integer time and discusses various representations including graphical, functional, tabular, and sequence representations. It then describes common elementary discrete time signals like impulse, step, ramp, and exponential signals. The document also classifies signals as energy/power signals, periodic/aperiodic, even/odd, and discusses basic manipulations. Finally, it defines discrete time systems as those that transform an input signal to an output signal according to a rule and classifies systems as static/dynamic and with finite/infinite memory.
![Discrete Time Systems
A discrete time system is a device or algorithm that performs a
prescribed operation on the discrete time signal, called input or
excitation, according to a well defined rules to produce another
discrete time signal called output or response of the system.
We say that the input signal 𝑥 𝑛 is transformed by the system in
to a signal y 𝑛 and express the general relationship between
𝑥 𝑛 and y 𝑛 as
y 𝑛 ≡ Τ[𝑥 𝑛 ]](https://image.slidesharecdn.com/dspclass3-200918005113/85/Dsp-class-3-34-320.jpg)
Dsp class 3
- 1. Discrete Time Signals and Systems -an Introduction
- 2. Discrete Time Signals The discrete time signal is a function of an independent variable that is an integer. It is important to note that discrete time signal is defined for integer values of time while it does not mean that at non integer instants the signal is ‘0’. It is just not defined over those intervals. The types of representation of signals are Graphical representation Functional representation Tabular representation Sequence representation
- 3. Contd.. Suppose we have 𝑥 𝑛 = 𝐶𝑜𝑠(𝜋𝑛) • Then 𝑥 𝑛 𝑎𝑡 𝑛 = 0 𝑚𝑒𝑎𝑛𝑠 𝐶𝑜𝑠 𝜋 × 0 =1 • 𝑥 𝑛 𝑎𝑡 𝑛 = 1𝑚𝑒𝑎𝑛𝑠𝐶𝑜𝑠 𝜋 × 1 = −1 • 𝑥 𝑛 𝑎𝑡 𝑛 = 2𝑚𝑒𝑎𝑛𝑠𝐶𝑜𝑠 𝜋 × 2 = 1 • 𝑥 𝑛 𝑎𝑡 𝑛 = 3𝑚𝑒𝑎𝑛𝑠𝐶𝑜𝑠 𝜋 × 3 = −1 . .
- 4. Graphical Representation of Discrete Time Signal
- 5. Functional Representation • 𝑥 𝑛 = 1, 𝑓𝑜𝑟 𝑛 = 1,3 4, 𝑓𝑜𝑟 𝑛 = 2 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
- 7. Sequence Representation Also called as four point sequence
- 8. Elementary Discrete Time Signals Unit Impulse Signal Unit Step Signal Unit Ramp Signal Exponential Signal
- 10. Unit step signal
- 11. Unit Ramp Signal
- 12. Exponential Signal 𝑥 𝑛 = 𝑎 𝑛 for all n We have two conditions here when a is Real, when a is complex. When a is real we have 4 conditions and those are explained in the graph. Let us take for example a is 2 that locates in a>1 then 𝑥 𝑛 = 2 𝑛 for all n
- 13. Contd..
- 14. Contd.. When a is complex valued then 𝑎 = 𝑟𝑒 𝑗𝜃 𝑥 𝑛 = (𝑟𝑒 𝑗𝜃) 𝑛 𝑥 𝑛 = 𝑟 𝑛 𝑒 𝑗𝑛𝜃 Then 𝑒 𝑗𝜃 = 𝐶𝑜𝑠𝜃 + 𝑗𝑆𝑖𝑛𝜃 𝑥 𝑛 = 𝑟 𝑛(𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑆𝑖𝑛𝑛𝜃) 𝑥 𝑛 = 𝑟 𝑛 𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃 When you have a+jb there ‘a’ is real and ‘b’ is imaginary. • 𝑥 𝑅 𝑛 = 𝑅𝑒(𝑟 𝑛 𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃) • 𝑥 𝑅 𝑛 = 𝑟 𝑛 𝐶𝑜𝑠𝑛𝜃 Real value signal • 𝑥𝐼 𝑛 = 𝐼𝑚(𝑟 𝑛 𝐶𝑜𝑠𝑛𝜃 + 𝑗𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃) • 𝑥𝐼 𝑛 = 𝑟 𝑛 𝑆𝑖𝑛𝑛𝜃 Imaginary signal
- 18. Classification of Discrete Time Signals In this section we classify discrete-time signals according to a number of different characteristics. The mathematical methods employed in the analysis of discrete-time signals and systems depend on the characteristics of the signals.
- 19. Energy Signals and Power Signals Energy Signal The Energy of a signal 𝑥(𝑛) is defined as 𝐸 = 𝑛=−∞ ∞ |𝑥 𝑛 |2 The magnitude square is considered since 𝑥(𝑛) can be real or complex. This will be useful to know whether the signal is having finite energy or infinite energy. In problems they will ask whether the signal is Energy Signal or not? Then you have to compute E if it is finite (0<E<∞) then it is Energy Signal.
- 20. Contd.. Power Signal Many Signals those posses infinite energy have finite average power. The average power of discrete time signal is 𝑃 = lim 𝑁→∞ 1 2𝑁+1 ( 𝑛=−𝑁 𝑁 |𝑥 𝑛 |2) Clearly if ‘E’ is finite. ‘P’=0 then Energy Signal. On the other hand if ‘E’ is infinite ‘P ‘may be either finite or infinite. If ‘P’ is finite (non zero) it is Power Signal. If ‘P’ is infinite it is neither Energy nor Power Signal.
- 21. Contd.. Examples • Unit Step is Energy Signal or Power Signal? Energy 𝐸 = 𝑛=−∞ ∞ |𝑥 𝑛 |2 • Unit Step Signal • 𝑥 𝑛 = 1 𝑓𝑜𝑟 𝑛 ≥ 0 0 𝑒𝑠𝑙𝑒 𝑤ℎ𝑒𝑟𝑒 Energy 𝐸 = 𝑛=0 ∞ |𝑥 𝑛 |2 • So 1+1+1…..∞ that is E= ∞ • When E is ∞ we have to verify for power signal.
- 22. Contd.. 𝑃 = lim 𝑁→∞ 1 2𝑁+1 ( 𝑛=−𝑁 𝑁 |𝑥 𝑛 |2) For unit step signal 𝑃 = lim 𝑁→∞ 1 2𝑁+1 ( 𝑛=0 𝑁 |𝑥 𝑛 |2 ) 𝑃 = lim 𝑁→∞ 1 2𝑁+1 (N+1) (if N=3 then 1+1+1+1=4…….) 𝑃 = lim 𝑁→∞ 1 2+ 1 𝑁 ( 1 𝑁 +1)= 1 2 Which is a finite value hence it is Power Signal.
- 23. Periodic and aperiodic Signals • 𝑥 𝑛 =𝑥 𝑛 + 𝑁
- 24. Even Signal • If the signal satisfies the condition 𝑥 −𝑛 =𝑥 𝑛 then it is an even signal
- 25. ODD Signal • If the signal satisfies the condition 𝑥 −𝑛 =−𝑥 𝑛 then it is an odd signal
- 26. Even or ODD? 𝑥1 𝑛 = 𝐶𝑜𝑠(0.125𝜋𝑛) To know whether the signal is even or odd? We have to compute 𝑥1 −𝑛 if it is equal to 𝑥1 𝑛 then it is even signal, if it is equal to −𝑥1 𝑛 then it is odd signal. So 𝑥1 −𝑛 = 𝐶𝑜𝑠 0.125𝜋 −𝑛 𝑥1 −𝑛 = 𝐶𝑜𝑠(−0.125𝜋(𝑛)) you know Cos(-𝜃)=Cos(𝜃) then 𝑥1 −𝑛 = 𝐶𝑜𝑠 0.125𝜋 𝑛 = 𝑥1 𝑛 So it is even signal Similarly if you do for 𝑥2 𝑛 you will identify it as odd signal. 𝑥2 𝑛 = 𝑆𝑖𝑛(0.125𝜋𝑛)
- 27. Simple Manipulations on Discrete Time Signals Transformation of the independent variable A signal x (n ) may be shifted in time by replacing the independent variable n by n-k, where k is an integer. If k is a positive integer, the time shift results in a delay of the signal by k units o f time. If k is a negative integer, the time shift results in an advance of the signal by |k| units in time
- 29. Folding of a signal • Another useful modification of the time base is to replace the independent variable n by -n. The result of this operation is a folding or a reflection of the signal about the time origin n =0.
- 30. How to plot 𝑥 −𝑛 + 2 ? • First do the folding and then shifting. • Note that because the signs o f n and k in 𝑥 𝑛 − 𝑘 and 𝑥 −𝑛 + 𝑘 are different, the result is a shift of the signals 𝑥 𝑛 and 𝑥 𝑛 − 𝑘 to the right by k samples, corresponding to a time delay.
- 33. Addition, multiplication and scaling of sequences Scaling Addition of two signals Product of two Signals
- 34. Discrete Time Systems A discrete time system is a device or algorithm that performs a prescribed operation on the discrete time signal, called input or excitation, according to a well defined rules to produce another discrete time signal called output or response of the system. We say that the input signal 𝑥 𝑛 is transformed by the system in to a signal y 𝑛 and express the general relationship between 𝑥 𝑛 and y 𝑛 as y 𝑛 ≡ Τ[𝑥 𝑛 ]
- 35. Input-Output description of the systems The input-output description of a discrete-time system consists of a mathe matical expression or a rule, which explicitly defines the relation between the input and output signals (input– output relationship). The exact internal structure of the system is either unknown or ignored. Examples 𝑦 𝑛 = 𝑥 𝑛 − 2 𝑦 𝑛 = 4𝑥(𝑛)
- 36. Classification of Discrete Time Systems Static vs Dynamic • A discrete-time system is called static or memory less if its output at any instant ‘n’ depends at most on the input sample at the same time, but not on past or future samples of the input. 𝑦 𝑛 = 𝑎𝑥(𝑛) 𝑦 𝑛 = 𝑛𝑥 𝑛 + 𝑏𝑥2 (𝑛) • In any other case, the system is said to be dynamic or to have memory . Finite Memory System Infinite Memory System