Post on 16-Jun-2018
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
15
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