Concept of frequency in Discrete Signals & Introduction to LTI Systems

Concept of frequency in Discrete Signals

Concept of frequency in Discrete Signals

Digital Filters

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

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,

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

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]

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.

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]

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]

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

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]).

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]

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

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

Properties of LTI Systems Causality

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

Stability

Properties of LTI Systems

Properties of LTI Systems

Properties of LTI Systems