Concept of frequency in Discrete Signals & Introduction to LTI Systems

Concept of frequency in Discrete Signals

&Introduction to LTI

Systems

Concept of frequency in Discrete Signals

Digital Filters

Fourier Series for continuous time periodic signals

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

dtetxjX tjaa

)()(

dejXtx tj

aa )(21)(

)()( jXtx aCTFT

a

Fourier Series for discrete time periodic signals

Fourier Transform Theorem & Properties

Discrete-Time Fourier Transform• Representation of a

sequence in terms of complex exponential sequence, ejωn

• The DTFT pair,

n

jj enxeX ][

deeXnx njj

)(21][

)(][ jF eXnx

Introduction to LTI System

• Discrete-time Systems– Function: to process a given input sequence

to generate an output sequence

Discrete-time systemx[n]

Input sequenc

e

y[n]Output

sequence

Fig: Example of a single-input, single-output 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 DTSx[n]

= αx1[n] + βx2[n]y[n]

= αy1[n] + βy2[n]

• ExampleIs 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.


• 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]


• 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]


• 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


• 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]).

Introduction to LTI System Impulse and Step Response

• 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 iny[n] = 0.5h[n+2] + 1.5h[n-1] - h[n-4]

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

k

knkxnx ][][][

kk

khknxknhkxny ][][][][][

][][][ nhnxny


• Properties of convolution– Commutative

– Associative

– Distributive

][][][][ 1221 nxnxnxnx

])[][(][][])[][( 321321 nxnxnxnxnxnx

][][][][])[][(][ 3121321 nxnxnxnxnxnxnx

• Causality

Properties of LTI Systems


• Stability– if and only if, sum of magnitude of Impulse

Response, h[n] is finite

n

nhS |][|

Stability

Concept of frequency in Discrete Signals & Introduction to LTI Systems

Documents

Transcript of Concept of frequency in Discrete Signals & Introduction to LTI Systems