1. Concept of frequency in Discrete Signals & Introduction to LTI Systems

2. Concept of frequency in Discrete Signals

3. Concept of frequency in Discrete Signals

4. Digital Filters

5. Digital Filters

6. Fourier Series for continuous time periodic signals

8. Fourier Transform Theorem & Properties Review of CTFT
Frequency domain representation of a continuous-time signal
The continuous-time signal xa(t) can be recovered from it’s CTFT, Xa(jΩ)
we denote the CTFT pair as

9. Fourier Series for discrete time periodic signals

10. Fourier Transform Theorem & Properties Discrete-Time Fourier Transform
Representation of a sequence in terms of complex exponential sequence, {ejωn}
The DTFT pair,

12. Introduction to LTI System
Discrete-time Systems
Function: to process a given input sequence to generate an output sequence
Discrete-time system
x[n]
Input sequence
y[n]
Output sequence
Fig: Example of a single-input, single-output system

13. Introduction to LTI System
Linear System
Most widely used
A Discrete-time system is a linear system if the superposition principle always hold.
If y1[n] and y2[n] are the response to the input sequences x1[n] and x2[n], then
Linear DTS
x[n]
= αx1[n] + βx2[n]
y[n]
= αy1[n] + βy2[n]

14. Introduction to LTI System
Example
Is the system described below linear or not ?
y[n] = x[n] + x[n-1]
Step :
a. Now, applying superposition by considering input as :
x[n] = ax[n] + bx[n]
b. Substitute the equation above with equation in (a), become
y[n] = (ax[n] + bx[n]) + (ax[n-1] + bx[n-1])
c. Rearrange the equation above become :-
y[n] = a(x[n] + x[n-1]) + b(x[n] + x[n-1]) => ay[n] + by[n]
c. The system is Linear since superposition is hold.

15. Introduction to LTI System
Shift-invariant System/Time-Invariant System
A shift (delay) in the input sequence cause a shift (shift) to the output sequence
If y1[n] is the response to an input x1[n], then the response to an input x[n] = x1[n - no] is y[n] = y1[n - no]

16. Introduction to LTI System
Causal System
Changes in output samples do not precede changes in input samples
y[no] depends only on x[n] for n ≤ no
Example:
y[n] = x[n]-x[n-1]

17. Introduction to LTI System
Stable System
For every bounded input, the output is also bounded (BIBO)
Is the y[n] is the response to x[n], and if
|x[n]| < Bx for all value of n
then
|y[n]| < By for all value of n
Where Bx and By are finite positive constant

18. Introduction to LTI System Impulse and Step Response
If the input to the DTS system is Unit Impulse (δ[n]), then output of the system will be
Impulse Response (h[n]).
If the input to the DTS system is Unit Step (μ[n]), then output of the system will be
Step Response (s[n]).

19. Introduction to LTI System Input-output Relationship
A Linear time-invariant system satisfied both the linearity and time invariance properties.
An LTI discrete-time system is characterized by its impulse response
Example:
x[n] = 0.5δ[n+2] + 1.5δ[n-1] - δ[n-4]
will result in
y[n] = 0.5h[n+2] + 1.5h[n-1] - h[n-4]

20. Introduction to LTI System Input-output Relationship
x[n] can be expressed in the form
where x[k] denotes the kth sample of sequence {x[n]}
The response to the LTI system is
or represented as

21. Introduction to LTI System Input-output Relationship
Properties of convolution
Commutative
Associative
Distributive

22. Properties of LTI Systems
Causality

23. Properties of LTI Systems
Stability
if and only if, sum of magnitude of Impulse Response, h[n] is finite

24. Stability

25. Properties of LTI Systems

26. Properties of LTI Systems

27. Properties of LTI Systems