4. DSP Theory III

Rahil Mahdian 18.05.2015

2

continuous DiscreteP

erio

dic

aPer

iodi

c

FS

CTFT DTFT

DFSDFT

FFT

Transforms - review

3

Convolution in DFT

DFT

In DFT convolutionproperty holds, but incyclic convolution sense.

Q. How to get Linearconvolution result ofLTI systems, usingCircular convolution?

4

(length N)

(length M)

Zeropadding(M-1) 0s

Zeropadding(N-1) 0s

N+M-1point DFT

N+M-1point DFT

N+M-1point IDFT

Linear Convolution from Circularprocessing in DFT world

5

Upsampling - interpolation

xi[n] in a low-pass filtered version of x[n]

The low-pass filter impulse response is

Hence the interpolated signal is written as

L/n

L/nsinnhi

ki

L/kLn

L/kLnsinkxnx

To create an upsampled signal, from the zeropaddedexpanded version, pass it through a LPF. (Gain something!)

6

Useful Noble Identities

M H(Z)

H(ZM) M

X[n]

X[n]

Xa[n]

Xb[n]

ya[n]

yb[n]

H(Z) LX[n] Xa[n] ya[n]

X[n] Xb[n] yb[n]L H(ZL)

7

Question

Problem. Explore the condition by which the changing the sequence ofupsampler and downsampler blocks, does not make a difference on the outputof the system?

M MN NX(n) X(n)y(n) y(n)

?

8

Polyphase filtering

=

Goal: Less number of multiplications per time unit Simpler structure of implementing a filter Used lso in filterbank

Refer to the Book, and Look at the Blackboard!

9

Ideal Filters

10

FIR filter - basics

No phase shift No distortion

Ideal delay filter

Linear phase shift is still good. (desirable)

11

FIR - constraints

Filter to have a limited number of the taps(components)

To have a linear phase shift Causal, stable and implementable As close as possible to the ideal desired filter

= k1+k2

12

FIR types (linear phase)

13

FIR filters

14

Butterworth Lowpass Filters

Passband is designed to be maximally flat

The magnitude-squared function is of the form

N2c

2

cj/j1

1jH

N2c

2

cj/s1

1sH

1-0,1,...,2Nkforej1s 1Nk2N2/jccN2/1

k

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Chebyshev Filters

Equiripple in the passband and monotonic in the stopband

Or equiripple in the stopband and monotonic in the passband

xcosNcosxV/V1

1jH 1N

c2N

2

2

c

16

Filter toolbox - MATLAB