01Introduction_Lecture8signalmoddiscr.pdf

Signals and Systems is a fundamental area of study in engineering and technology that deals with the analysis and processing of signals and the behavior of systems that handle these signals.

Signals
A signal is a time-varying quantity that conveys information. Signals can be in various forms, such as electrical voltages, sound waves, or digital data. They are typically classified into two main types:

Analog Signals: Continuous signals that vary smoothly over time.
Digital Signals: Discrete signals that change in distinct steps or levels.
Systems
A system is a collection of components or modules that process input signals to produce output signals. Systems can be described in terms of their response to input signals and are often categorized as:

Linear Systems: Systems where the principle of superposition applies.
Non-linear Systems: Systems where the output is not directly proportional to the input.
Key Concepts
Time and Frequency Domains: Signals and systems can be analyzed in both time and frequency domains. The Fourier transform is a key tool for converting signals between these domains.
Filtering: The process of removing unwanted components from a signal.
Modulation: The technique of varying a signal to encode information.
Sampling: The process of converting a continuous signal into a discrete signal.
Applications
Signals and systems are crucial in various fields, including:

Communications: For transmitting data over distances.
Control Systems: For regulating the behavior of dynamic systems.
Signal Processing: For enhancing or extracting information from signals.
Electronics: For designing circuits and devices that interact with signals .

1. Lecture 8 Introduction to signals and systems Continued
Basic signals models - DISCRETE TIME
 Unit Impulse functions δ[n]
 It is defined as ??????[�]=[{
1 �=0
0 �≠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]

3.  Unit step functions u[n]
u[n] = {
1 �≥0
0 &#3627408475;<0
and delayed step u[n-2]


Considering arbitrary everlasting signal, 0.8
n
cos2πn,
Using step function, 0.8
n
cos2πn u[n] and
0.8
n
cos2πn {u[n] –u[n-12] }

4. Using step function constructing a unit delayed discrete pulse. u[n-1] – u[n-3]



 The exponential functions

5. o x[n] = γ
n
where γ is complex in general given by
γ =e
λ
= Ε +jΩ = r θ = r e
jθ

o (e
λ
)
n
actually expressed as γ
n
where γ = e
λ
OR λ = ln γ
o e
-0.3 n
actually expressed as (0.7408)
n
; where e
-0.3
= 0.7408
Now γ
n
= (e
λ
)
n
; If λ = ε +jω ;
then γ
n
= (e
ε +jω
)
n
; = e
εn
e
jωn
= r
n
e
jωn
= r
n
[cos ωn+ jsin ωn ]

o γ
n
compasses Large class of functions : viz (Here , λ = ε +jω and γ = e
λ
)
1. λ = 0 i.e. γ = e
0
= 1 a constant K = K e
0n
= 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 = e
ε
; 4. λ = ε +jω A exponentially varying sinusoid (e
ε
)
n
[cos ωn ]

6. Ex1: (0.7408)
n
; where e
-0.3
= 0.7408 Ex: (0.7408)
n
[cos3πn] = e
-0.3n
[cos3πn]
Ex2: (4)
n
; where e
1.386
= 4 e.t.c.
A monotonic exponential (e
ε
)
n
ε is +ve or –ve real values





Salient Points :

7. 1) If λ lies in the imaginary axis, the corresponding γ (= e
ε
) lies on a unit circle with centre origin of the complex plane
2) γ
n
= (e
ε +jω
)
n
; = e
εn
e
jωn
= r
n
[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

8. +

9. 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 =
??????
2??????
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
??????
??????
cycles per sample
Eg: ??????&#3627408476;??????(
??????
12
&#3627408475;+
??????
4
) Ω =
??????
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 =
&#3627408474;??????
??????
[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 = ??????
&#3627408474;??????
??????
=
&#3627408474;
??????
2?????? OR
Period of the discrete time signal is No =
2&#3627408474;??????
??????

 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
2??????
is rational. This means that the frequency
F (cycles per samples) should be in the form of ratio of two integers”

10.  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 e
jΩn
)
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
2??????
??????
with m as the smallest integer to make Period No integer.
Example: cos
3??????
7
&#3627408475; here,
??????
2??????
=
3
14
; with m = 3 , No = 3 Χ
14
3
= 14


Illustration of periodicity and aperiodicity of the discrete sinor signal

11. 5) Allowable unique variation to discrete time frequency Ω =
2??????
??????
(radian frequency) is finite and =0 to 2π. The rule is ΩN = 2π
Additional readings to support this statement: cos
3??????
7
&#3627408475;=??????&#3627408476;??????((
3??????
7
+2??????)&#3627408475;)=??????&#3627408476;??????
17??????
7
&#3627408475;

13. 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)
(E
N
+a1E
N-1
+ a2E
N-2
+….+aN-1E+aN)y(n) = (boE
m
+b1E
m-1
+ b2E
m-2
+….+bM-1E +bM)x(n) OR
Q(E) y(n) = P(E) x(n)
 Transfer function model
&#3627408486;(&#3627408475;)
&#3627408485;(&#3627408475;)
=
b0E
m
+b1E
m−1
+ b2E
m−2
+⋯+bm−1E+bm
E
n
+a1E
n−1
+ a2E
n−2
+⋯+an−1E+an
OR
??????(??????)=
&#3627408486;[&#3627408475;]
&#3627408485;[&#3627408475;]
=
&#3627408451;[??????]
&#3627408452;[??????]


 Frequency response model OR Sinusoidal transfer function model
??????(&#3627408466;
&#3627408471;??????
)=
&#3627408451;[(&#3627408466;
&#3627408471;??????
]
&#3627408452;[(&#3627408466;
&#3627408471;??????
]


 State space model

&#3627408485;[&#3627408475;+1]=&#3627408467;[&#3627408485;,??????,&#3627408475;] ----------> State equation
&#3627408486;=&#3627408468;[&#3627408485;,??????,&#3627408475;] -----------> Output equation

Where, &#3627408485; State variables ; ?????? Input signal ; &#3627408467; and &#3627408468; are functions

14. 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
??????
{&#3627408485;[&#3627408475;]−&#3627408485;[&#3627408475;−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] = ∑ &#3627408485;[??????]
&#3627408475;
&#3627408472;=−∞ -------- 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

15. SOME comparisons among Continuous and discrete time signals