01Introduction_Lecture8signalmoddiscr.pdf

Lecture 8 Introduction to signals and systems Continued
Basic signals models - DISCRETE TIME
 Unit Impulse functions δ[n]
 It is defined as 𝛿[𝑛] = [{
1 𝑛 = 0
0 𝑛 ≠ 0
It is a deterministic signal unlike continuous time equivalent
 Multiplication of a function by an impulse
 x[n] δ[n] = x[0] δ[n] is an impulse of strength x[0] at n= 0
 x[n] δ[n-T] = x[T] δ[n-T] is an impulse of strength x[T] at n= T
 Sampling property of the unit impulse function
 Any general sequence x[n] = {5,3, 2
↑ ,5,3,7,5} can be represented as
x[n] = 5 δ[n+2] +3 δ[n+1] + 2 δ[n] + 5 δ[n-1] +3 δ[n-2] +7 δ[n-3] +5 δ[n-4]
 Impulse function does define a unique function
 It is a true function in ordinary sense
 Its range is defined and is symmetric about n = 0. i.e. δ[n] = δ[-n]

 Unit step functions u[n]
u[n] = {
1 𝑛 ≥ 0
0 𝑛 < 0
and delayed step u[n-2]
Considering arbitrary everlasting signal, 0.8n
cos2πn,
Using step function, 0.8n
cos2πn u[n] and
0.8n
cos2πn {u[n] –u[n-12] }

Using step function constructing a unit delayed discrete pulse. u[n-1] – u[n-3]
 The exponential functions

o x[n] = γn
where γ is complex in general given by
γ =eλ
= Ε +jΩ = r θ = r ejθ
o (eλ
)n
actually expressed as γn
where γ = eλ
OR λ = ln γ
o e-0.3 n
actually expressed as (0.7408)n
; where e-0.3
= 0.7408
Now γn
= (eλ
)n
; If λ = ε +jω ;
then γn
= (eε +jω
)n
; = eεn
ejωn
= rn
ejωn
= rn
[cos ωn+ jsin ωn ]
o γn
compasses Large class of functions : viz (Here , λ = ε +jω and γ = eλ
)
1. λ = 0 i.e. γ = e0
= 1 a constant K = K e0n
= K(1)n
2. λ = jω i.e. with ε = 0 A Sinusoid : cos ωn , cos 5n, cos2πn
3. λ = ε i.e. with ω= 0 ; i.e. γ = eε
; 4. λ = ε +jω A exponentially varying sinusoid (eε
)n
[cos ωn ]

Ex1: (0.7408)n
; where e-0.3
= 0.7408 Ex: (0.7408)n
[cos3πn] = e-0.3n
[cos3πn]
Ex2: (4)n
; where e1.386
= 4 e.t.c.
A monotonic exponential (eε
)n
ε is +ve or –ve real values
Salient Points :

1) If λ lies in the imaginary axis, the corresponding γ (= eε
) lies on a unit circle with centre origin of the complex plane
2) γn
= (eε +jω
)n
; = eεn
ejωn
= rn
[cos ωn+ jsin ωn ]
If ε = 0 ; with different value of ω, it is in unit circle.
If ε > 0 ; with different value of ω, it is in exterior of unit circle.
If ε < 0 ; with different value of ω, it is in interior of unit circle.
The λ – plane, the γ – plane , and their mapping

3) Discrete time sinusoid C cos(Ωn+θ)
Where, C is the amplitude θ is the phase in radians
Ω is the discrete frequency in radians per sample and n is the integer values (or discrete time variable)
Ω is also called (radian frequency) -----> radians / sample and Ωn is angle in radians ;
Let C cos(Ωn+θ) = C cos(2πFn+θ) Where, F =
𝛺
2𝜋
is the discrete time frequency
i.e ( radians /2π ) per sample OR cycles per sample
If No is the period (sample / cycle) of the sinusoid, then the frequency of the sinusoid F =
1
𝑁𝑜
cycles per sample
Eg: 𝑐𝑜𝑠 (
𝜋
12
𝑛 +
𝜋
4
) Ω =
𝜋
12
radians per sample
F =
1
24
cycles per sample OR No = 24 samples per cycle
4)Discrete time sinusoid may be periodic or non-periodic in nature.
 Discrete time signal is periodic only when the sampling interval T is the continuous time period τ multiplied by a rational number.
OR T =
𝑚𝜏
𝑁
[s / sample]
where, two integers m and N are relatively prime (coprime) i.e. they have no common devisor except 1. Then,
Discrete time radian frequency Ω = ω T = 𝜔
𝑚𝜏
𝑁
=
𝑚
𝑁
2𝜋 OR
Period of the discrete time signal is No =
2𝑚𝜋
𝜔
 In another perspective (using discrete time frequency F), a condition for periodicity of a discrete time signal is also defined as:
“a discrete time sinusoidal signal is periodic only if its frequency F =
Ω
2𝜋
is rational. This means that the frequency
F (cycles per samples) should be in the form of ratio of two integers”

 In a continuous time sinusoid signal, Cos ω t is periodic, regardless of the value of ω. i.e. for all ω = 1.1 , 3.6 , 4.5π e.t.c.
 Such is not the case for discrete – time sinusoid Cos Ω n (or exponential ejΩn
)
i.e., for all Ω = 1.2 , 3.6 , 4.5 e.t.c. the discrete time signals are not periodic and
for all Ω = 1.2π , 3.6π , 4.5π e.t.c. the discrete time signals are periodic
 Then periodicity can be calculated by No = m
2𝜋
𝛺
with m as the smallest integer to make Period No integer.
Example: cos
3𝜋
7
𝑛 here,
𝛺
2𝜋
=
3
14
; with m = 3 , No = 3 Χ
14
3
= 14
Illustration of periodicity and aperiodicity of the discrete sinor signal

5) Allowable unique variation to discrete time frequency Ω =
2𝜋
𝑁
(radian frequency) is finite and =0 to 2π. The rule is ΩN = 2π
Additional readings to support this statement: cos
3𝜋
7
𝑛 = 𝑐𝑜𝑠 ((
3𝜋
7
+ 2𝜋) 𝑛) = 𝑐𝑜𝑠
17𝜋
7
𝑛

System models: (input-output description)
 Difference equation
A Discrete Time system represented by a linear difference equation as
y(n+N) + a1y(n+N-1) + a2y(n+N-2) +….+ aN-1y(n+1) + any(n) = bo x(n+M) + b1x(n+M-1) + b2x(n+M-2) +….+ bM-1 x(n+1) + bMx(n)
(EN
+a1EN-1
+ a2EN-2
+….+aN-1E+aN)y(n) = (boEm
+b1Em-1
+ b2Em-2
+….+bM-1E +bM)x(n) OR
Q(E) y(n) = P(E) x(n)
 Transfer function model
𝑦(𝑛)
𝑥(𝑛)
=
b0Em+b1Em−1+ b2Em−2+⋯+bm−1E+bm
En+a1En−1+ a2En−2+⋯+an−1E+an
OR
𝐺(𝐸) =
𝑦[𝑛]
𝑥[𝑛]
=
𝑃[𝐸]
𝑄[𝐸]
 Frequency response model OR Sinusoidal transfer function model
𝐺(𝑒𝑗𝜔
) =
𝑃[(𝑒𝑗𝛺
]
𝑄[(𝑒𝑗𝛺]
 State space model
𝑥[𝑛 + 1] = 𝑓[𝑥, 𝑢, 𝑛] ----------> State equation
𝑦 = 𝑔[𝑥, 𝑢, 𝑛] -----------> Output equation
Where, 𝑥 State variables ; 𝑢 Input signal ; 𝑓 and 𝑔 are functions

DISCRETE SYSTEM CLASSIFICATION IS A SIMILAR DISCUSSION AS COMPARED TO CONTINIOUS TIME SYSTEMS
Discrete time system examples :
Example : Digital differentiator
The output y(t) is required to be the derivative of the input x(t)
Y[n]= =
1
𝑇
{𝑥[𝑛] − 𝑥[𝑛 − 1]}
The sampling interval be sufficiently small
Digital integrator
The output y(t) is required to be the integration of the input x(t)
Y[n] = ∑ 𝑥[𝑘]
𝑛
𝑘=−∞ -------- 1)
Y[n] - y[n-1] = T x[n] ----------2)
In eqn. 1), the output y[n] at any instant n is computed by adding all past values of input till n ----> Non recursive form.
In eqn. 2), the computation of y[n] involves addition of only two values, preceding output value and present input value
--------> Recursive form

SOME comparisons among Continuous and discrete time signals

01Introduction_Lecture8signalmoddiscr.pdf

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