adsp lec 02

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Dr. Tahir Zaidi

Advanced Digital Signal Processing

Lecture 2

Signal Representation and Time Domain Analysis

Basic Types of Digital Signals Basic Types of Digital Signals

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Basic Types of Digital Signals

Sine and Exp Using Matlab

% sine generation: A*sin(omega*n+theta)

% exponential generation: A^n

n = 0: 1: 50;

% amplitude

A = 0.87;

% phase

theta = 0.4;

% frequency

omega = 2*pi / 20;

% sin generation

xn1 = A*sin(omega*n+theta);

% exp generation

xn2 = A.^n;

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Basic Operations

Operations in Matlab

xn1 = [1 0 3 2 -1 0 0 0 0 0];

xn2 = [1 3 -1 1 0 0 1 2 0 0];

yn = xn1 + xn2;

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x[n] via impulse functions

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Input: sum of weighted shifted impulses

Time Domain Analysis

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Linear Time-Invariant Systems

Linear

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Linear Time-Invariant Systems Linear Time-Invariant System

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Linear Time-Invariant System

Input: sum of weighted shifted impulses

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Using Linearity and Time-Invariance for the impulses

Sum of wt. Shifted impulses – sum of wt. Shifted impulse responses

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LTI System

Two ways

As the representation of the output as a sum of delayed and scaled impulse responses.

As a computational formula for computing y[n] (“y at time n”) from the entire sequences x and h.

Form x[k]h[n-k] for -∞<k<+∞ for a fixed n

Sum over all k to produce y[n]

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Convolution in the time domain: [ ] [ ] [ ]k

y n x k h n k

y[n] = 2 –3 3 3 –6 0 1 0 0

Example-Convolution of Two Rectangles

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Example..(Continued)

Example-Convolution Of Two Sequences

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Stability

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Causality

Causality & Stability- Example

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Difference Equation

For all computationally realizable LTI systems, the input and output satisfy a difference equation of the form

This leads to the recurrence formula

which can be used to compute the “present” output from the present and M past values of the input and N past values of the output

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Linear Constant-Coefficient Difference(LCCD) Equations

Linear Constant-Coefficient Difference (LCCD) Equations…( Continued)

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Linear Constant-Coefficient Difference (LCCD) Equations….( Continued)

First-Order Example

Consider the difference equation

y[n] =ay[n−1] +x[n]

We can represent this system by the following block diagram:

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Exponential Impulse Response

With initial rest conditions, the difference Equation has impulse response

y[n] =ay[n−1] +x[n]

h[n] =anu[n]

Linear Constant-Coefficient Difference (LCCD) Equations….( Continued)

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Digital Filter

Y = FILTER(B,A,X)

filters the data in vector X with the filter described by

vectors A and B to create the filtered data Y. The filter

is a "Direct Form II Transposed" implementation of the

standard difference equation:

a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... +

b(nb+1)*x(n-nb) - a(2)*y(n-1) - ... - a(na+1)*y(n-na)

[Y,Zf] = FILTER(B,A,X,Zi)

gives access to initial and final conditions, Zi and Zf, of

the delays.

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LTI summary

Complex Exp Input Signal

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The Frequency Response

Delay and First Difference

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More on the Ideal Delay

Discrete Time Fourier Transform

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Properties

Properties…(cont)

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Symmetric Properties

DTFT Properties:

x[n] X(e j )

x*[n] X*(e j )

x[n] x[n] X(e j ) X(e j )

even even

x[n] x[n] X(e j ) X(e j )

odd odd

x[n] x*[n] X(e j ) X*(e j )

real Hermitian symmetric

Consequences of Hermitian Symmetry

If

then

And

X(e j ) X*(e j )

Re[X(e j )] is even

Im[X(e j )] is odd

X(e j ) is even

X(e j ) is odd

If x[n] is real and even, X(e j ) will be real and even

and if x[n] is real and odd, X(e j ) will be imaginary and odd

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DTFT- Sinusoids

DTFT of Unit Impulse

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Ideal Lowpass Filter

Example

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Magnitude and Angle Form

Magnitude and Angle Plot

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Example

Real and Even ( Zero Phase)

Consider an LTI system with an even unit sample response

DTFT is

e 2 j + 2 e

j + 3 + 2 e j + e

2 j

2 cos( 2 ) + 4 cos( ) + 3

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Real & Even (Zero Phase)

Frequency response is real, so system has “zero” phase shift

This is to be expected since unit sample response is real and even.

Linear Phase

H(z) e2 j + 2e j + 3+3e j + e2 j

e2 j (e2 j + 2e j + 3+2e j + e2 j )

e2 j (2cos(2 )+ 4cos( )+3)

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Useful DTFT Pairs

Convolution Theorem

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Linear Phase… ( cont.)

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Frequency Response of DE

Matlab

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Example

Ideal Filters

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Ideal Filters

HP Digital Filter from LP Design

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BP Digital Filter from HP & LP

Ideal Lowpass Filter

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h[n] of ideal filter

Approximations

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Freq Axis

Inverse System