Digital signal processing 22 scheme notes

Discrete Fourier Transform
1
• The DFT pair was given
as N  1 1
N
N  1
 
j 2  / N
kn
X k 
 n 
0
 x[n]e
 j 2  / N kn x[n]  
k  0 X k
e
• Baseline for computational
complexity:
– Each DFT coefficient requires
• N complex multiplications
• N-1 complex additions
– All N DFT coefficients require
• N2 complex multiplications
• N(N-1) complex additions
• Complexity in terms of real operations
• 4N2 real multiplications
• 2N(N-1) real additions
• Most fast methods are based on symmetry
properties
– Conjugate symmetry
– Periodicity in n and k
e  j2  / N k N  n 
 e  j2  / N kN
e  j2  / N k  n   e j2  / N kn
e  j2  / N kn
 e  j2  / N k n  N 
 e j2  / N k  N n

6
2
rk
N
N
rk
N / 2
N
Decimation-In-Time FFT Algorithms
• Makes use of both symmetry and periodicity
• Consider special case of N an integer power of
2
• Separate x[n] into two sequence of length N/2
– Even indexed samples in the first sequence
– Odd indexed samples in the other sequence
X k
 
N  1
 x[n]e
n  0
 j2  / N kn
N  1

x[n
]e
n
even

 j2  / N kn
N  1
 x[n]e
n odd

 j 2  / N kn
• Substitute variables n=2r for n even and n=2r+1 for
odd N / 2  1
X k
 

r  0
x[2
r ]W
 1]W
2 r  1 k
N / 2  1
 
x[2 r
r  0
N / 2
 1
N / 2  1
 
r  0
x[2
r ]W
N
r  0
N / 2
 W k
 x[2 r  1]W rk
k
 G k   W H k

• G[k] and H[k] are the N/2-point DFT’s of each
subsequence

Decimation In Time
• 8-point DFT example using
decimation-in-time
• Two N/2-point DFTs
– 2(N/2)2 complex
multiplications
– 2(N/2)2 complex additions
• Combining the DFT
outputs
– N complex
multiplications
– N complex additions
• Total complexity
– N2/2+N complex
multiplications
– N2/2+N complex
additions
– More efficient than direct
DFT
• Repeat same process
– Divide N/2-point DFTs
into
7

Decimation In Time Cont’d
• After two steps of decimation in
time
• Repeat until we’re left with two-point
DFT’s
8

Decimation-In-Time FFT Algorithm
• Final flow graph for 8-point decimation in
time
• Complexity:
– Nlog2N complex multiplications and
additions
9

Butterfly Computation
• Flow graph constitutes of
butterflies
• We can implement each butterfly with one multiplication
• Final complexity for decimation-in-time FFT
– (N/2)log2N complex multiplications and
additions
10

11
In-Place Computation
• Decimation-in-time flow graphs require two sets of
registers
– Input and output for each stage
• Note the arrangement of the input indices
– Bit reversed indexing
X 0 0

 x 0

 X 0 000  x
000

X 0 1

 x 4

 X 0 001  x
100

X 0 2

 x 2

 X 0 010  x
010

X 0 3

 x 6

 X 0 011  x
110

X 0 4

 x 1

 X 0 100  x
001

X 0 5

 x 5

 X 0 101  x
101


12
Decimation-In-Frequency FFT Algorithm
nk
N
N N N
n 2
r
N
N N / 2
N / 2
X k
 
• The DFT equation
N  1

n  0
• Split the DFT equation into even and odd frequency
indexes
N  1 N / 2  1 N  1
x[n]W
X 2 r
 
 x[n]W n 2 r

x[n]W n 2 r 


n  0
x[n]W n 2 r
n  0
• Substitute variables to get
n  N / 2
N / 2  1 N / 2  1
  x[n
n  0
N / 2  1
X 2 r
 

n 
0
x[n]W  N / 2 ]W
n  N / 2 2 r  
n 
0
x[n]  x[n  N / 2 ]W nr
• Similarly for odd-numbered frequencies
N / 2  1
X 2 r  1    x[n]  x[n
 N / 2 ]W n 2 r  1 
n 
0

Decimation-In-Frequency FFT Algorithm
• Final flow graph for 8-point decimation in
frequency
13

x n  y n

D/C
H(ej)
C/D
He j   H j / T 
c
Filter Design Techniques
• Any discrete-time system that modifies certain frequencies
• Frequency-selective filters pass only certain frequencies
• Filter Design Steps
– Specification
• Problem or application specific
– Approximation of specification with a discrete-time system
• Our focus is to go from spec to discrete-time system
– Implementation
• Realization of discrete-time systems depends on target technology
• We already studied the use of discrete-time systems to implement a
continuous-time system
– If our specifications are given in continuous time we can use
xc(t)
yr(t)
 

14
IIR Filters

Digital Filter Specifications
• Only the magnitude approximation problem
• Four basic types of ideal filters with magnitude
responses as shown below (Piecewise flat)
15

16
• These filters are unealisable because (one of the
following is sufficient)
– their impulse responses infinitely long non-
causal
– Their amplitude responses cannot be equal to
a constant over a band of frequencies
Another perspective that provides some
understanding can be obtained by looking at
the ideal amplitude squared.

17
• The realisable squared amplitude response
transfer function (and its differential) is
continuous in
• Such functions 
– if IIR can be infinite at point but around
that point cannot be zero.
– if FIR cannot be infinite anywhere.
• Hence previous differential of ideal response
is unrealisable

• For example the magnitude response of a
digital lowpass filter may be given as indicated
below
18

p
• In the passband 0     we
requireG ( e j ) 
1
that with a deviation
 p
p
1    G ( e j )  1   p ,   
p
  

• In the stopband  s we re qs
uire G ( e
j
)  0
that with a deviation
G ( e j )   s ,  s   

19

p
Filter specification parameters


s

p

s
• - passband edge frequency
• - stopband edge frequency
• - peak ripple value in the
passband
• - peak ripple value in the
stopband
20

• Practical specifications are often given
in terms of loss function (in dB)
• G ( )

 20 log 10 G ( e
j )
• Peak passband
ripple p   20 log
 10 (1   p ) dB
• Minimum stopband attenuation
 s   20
log 10 ( s ) dB
21

p
s
p
• In practice, passband edge frequency F
and
stopband edge frequency arFes
specified in Hz
• For digital filter design, normalized
bandedge frequencies need to be computed
from
specifications in Hz using  2
F
p
 
 FT
p
FT
p
 2 FT
2
Fs

s
 s
   2 F
T
FT FT
250

p s
T
 7
• Example - Let F
kHz,
F  3
F  25 kHz
kHz,
and
• Then

2( 7  10 3 )
p  3  0 .56

25  10
2( 3  10 3 )
 s  3  0 .24

25  10
23

IIR Digital Filter Design
24
Standard approach
(1)Convert the digital filter specifications into
an analogue prototype lowpass filter
specifications
(2)Determine the analogue lowpass filter
transfer function H a ( s )
(3)Transform H a ( s ) by replacing the
complex variable to the digital transfer
function
G ( z )

25
• This approach has been widely used for
the following reasons:
(1)Analogue approximation techniques
are highly advanced
(2)They usually yield closed-
form solutions
(3)Extensive tables are available
for analogue filter design
(4)Very often applications require
digital simulation of analogue systems

D a ( s )
• Let an analogue transfer function
be H ( s )

a
Pa
( s )
where the subscript “a” indicates
the analogue domain
• A digital transfer function derived from
this is denoted as
G ( z )

P
( z )
D
( z )
26

27
• Basic idea behind the conversion of H ( s ) intoG
( z )
a
is to apply a mapping from the s-
domain to the z-
domain so that essential properties of the
analogue frequency response are preserved
• Thus mapping function should be such that
– Imaginary (j  ) axis in the s-plane
be mapped onto the unit circle of the
z-plane
– A stable analogue transfer function be
mapped into a stable digital transfer function

Specification for effective frequency response of a continuous-time
lowpass filter and its corresponding specifications for discrete-time
system.
dp or d1 passband ripple
ds or d2 stopband ripple
Wp, wp passband edge frequency
Ws, ws stopband edge frequency
e2 passband ripple parameter
1 – dp = 1/1 + e2
BW bandwidth = wu – wl
wc 3-dB cutoff frequency
wu, wl upper and lower 3-dB cutoff
frequensies
Dw transition band = |wp – ws|
Ap passband ripple in dB
=  20log10(1 
dp) As stopband attenuation in
dB
= -
20log10(ds) 28

29
Design of Discrete-Time IIR Filters
• From Analog (Continuous-Time)
Filters
– Approximation of Derivatives
– Impulse Invariance
– the Bilinear Transformation

Reasons of Design of Discrete-Time IIR Filters
from Continuous-Time Filters
30
• The art of continuous-time IIR filter design is highly advanced and,
since useful results can be achieved, it is advantageous to
use the design procedures already developed for continuous-time
filters.
• Many useful continuous-time IIR design methods have relatively
simple closed-form design formulas. Therefore, discrete-time
IIR filter design methods based on such standard continuous-
time design formulas are rather simple to carry out.
• The standard approximation methods that work well for continuous-
time IIR filters do not lead to simple closed-form design
formulas when these methods are applied directly to the
discrete-time IIR case.

31
Characteristics of Commonly Used Analog Filters
• Butterworth Filter
• Chebyshev Filter
– Chebyshev Type I
– Chebyshev Type II of Inverse Chebyshev Filter

32
Butterworth Filter
• Lowpass Butterworth filters are all-pole filters characterized by the magnitude-squared
frequency response
|H(W)|2 = 1/[1 + (W/Wc)2N] = 1/[1 + e2(W/Wp)2N]
where N is the order of the filter, Wc is its – 3-dB frequency (cutoff frequency), Wp
is the bandpass edge frequency, and 1/(1 + e2) is the band-edge value of |
H(W)|2.
• At W = Ws (where Ws is the stopband edge frequency) we have
1/[1 + e2(Ws/Wp)2N] = d2
2
and
N = (1/2)log10[(1/d2
2) – 1]/log10(Ws/Wc) = log10(d/e)/log10(Ws/Wp)
where d2= 1/1 + d2
2.
• Thus the Butterworth filter is completely characterized by the parameters N, d2, e, and
the ratio Ws/Wp.

2
H j  c
Butterworth Lowpass
Filters
• Passband is designed to be maximally flat
• The magnitude-squared function is of the
form 
1
c
1  j / j
2 N
H s 
2

1  s / j
1
c
2 N
sk
  1 1 / 2 N
j    e
 j / 2 N 2 k  N  1 
c c
for k  0,1,...,2N - 1
c
33

Frequency response of lowpass Butterworth
filters
34

35
Chebyshev Filters
• The magnitude squared response of the analog lowpass Type I
Chebyshev filter of Nth order is given by:
|H(W)|2 = 1/[1 + e2T 2(W/W )].
N
where TN(W) is the Chebyshev polynomial of order
N: TN(W) = cos(Ncos-1 W), |W|  1,
= cosh(Ncosh-1 W), |W| >
1.
The polynomial can be derived via a recurrence relation given
by Tr(W) = 2WTr-1(W) – Tr-2(W), r  2,
with T0(W) = 1 and T1(W) = W.
• The magnitude squared response of the analog lowpass Type II or
inverse Chebyshev filter of Nth order is given by:
|H(W)|2 = 1/[1 + e2{TN(Ws/Wp)/ TN(Ws/W)}2].

c
N
N
c
Chebyshev
Filters
• Equiripple in the passband and monotonic in the stopband
• Or equiripple in the stopband and monotonic in the
passband
2 1

H  j 
1   2
V 2
 /  
V x   cos N cos  1 x 
36

N
Frequency response of
lowpass Type I Chebyshev
filter
|H(W)|2 = 1/[1 + e2T 2(W/W )]
Frequency response of
lowpass Type II Chebyshev filter
|H(W)|2 = 1/[1 + e2{T 2(W /W )/T 2(W /W)}]
N
s
p
N
s
37

38
N = log10[( 1 - d 2 +  1 – d 2(1 + e2))/ed
]/log 2 2 2
10
[(W /W ) +  (W /W )2 –
1 ]
s p s p
= [cosh-1(d/e)]/[cosh-1(Ws/Wp)]
for both Type I and II Chebyshev filters, and
where d2 = 1/  1 + d2.
• The poles of a Type I Chebyshev filter lie on an ellipse in the s-plane with major
axis r1 = Wp{(b2 + 1)/2b] and minor axis r1 = Wp{(b2 - 1)/2b] where b is related to
e according to
b = {[ 1 + e2 + 1]/e}1/N
• The zeros of a Type II Chebyshev filter are located on the imaginary axis.

Type I: pole positions are
xk = r2cosfk
yk = r1sinfk
fk = [p/2] + [(2k + 1)p/2N]
r1 = Wp[b2 + 1]/2b
r2 = Wp[b2 – 1]/2b
b = {[ 1 + e2 + 1]/e}1/N
Type II: zero positions are
sk = jWs/sinfk
and pole positions are
vk = Wsxk/ x 2 + y 2
k
k
wk = Wsyk/ x 2 + y 2
k
k 2
b = {[1 +  1 – d 2
]/d }1/N
Determination of the pole locations
for a Chebyshev filter.
k = 0, 1, …, N-1.
39

Approximation of Derivative Method
• Approximation of derivative method is the simplest one for converting an
analog filter into a digital filter by approximating the differential equation
by an equivalent difference equation.
– For the derivative dy(t)/dt at time t = nT, we substitute the backward difference
[y(nT) – y(nT – T)]/T. Thus

t  nT
 T )

y [ n ]  y [ n  1]
T
where T represents the sampling period. Then, s = (1 – z-1)/T
– The second derivative d2y(t)/dt2 is derived into second difference as follow:

t  nT T 2
y [ n ]  2 y [ n  1]  y [ n  2 ]
which s2 = [(1 – z-1)/T]2. So, for the kth derivative of y(t), sk = [(1 – z-1)/T]k.
dy ( t ) y ( nT )  y ( nT
dt T
dy
( t )
dt
40

Approximation of Derivative
Method
41
• Hence, the system function for the digital IIR filter obtained as a result of
the approximation of the derivatives by finite difference is
H(z) = Ha(s)|s=(z-1)/Tz
• It is clear that points in the LHP of the s-plane are mapped into the
corresponding points inside the unit circle in the z-plane and points in
the RHP of the s-plane are mapped into points outside this circle.
– Consequently, a stable analog filter is transformed into a stable digital filter due
to this mapping property.
jW
Unit circle
s
s-
plane
z-plane

Example: Approximation of derivative method
Convert the analog bandpass filter with system function
Ha(s) = 1/[(s + 0.1)2 + 9]
Into a digital IIR filter by use of the backward difference for the
derivative.
Substitute for s = (1 – z-1)/T into Ha(s) yields
H(z) = 1/[((1 – z-1)/T) + 0.1)2
+ 9]
H ( z ) 
T 2
1  0 .2 T  9 .01 T 2
 1
1 
2 ( 1  0 .1 T ) z
 1
z
 2
1  0 .2 T  9 .01 T 2
1  0 .2 T  9 .01 T 2
T can be selected to satisfied specification of designed filter. For example, if T =
0.1, the poles are located at
p1,2
= 0.91  j0.27 = 0.949exp[
j16.5o]
42

• If the continuous-time filter is bandlimited to
H c
 j   0
c 
Filter Design by Impulse Invariance
• Remember impulse invariance
– Mapping a continuous-time impulse response to discrete-time
– Mapping a continuous-time frequency response to discrete-time
hn  
T h
nT

d
c
d
   2

H e j
   H  j  j
c
k 

 Td
Td

k


j

 

   /
T d
H e   H 
j


Td

 

• If we start from discrete-time specifications Td cancels out
– Start with discrete-time spec in terms of 
– Go to continuous-time =  /T and design continuous-time
filter
– Use impulse invariance to map it back to discrete-time = T
• Works best for bandlimited filters due to possible aliasing
43

Impulse Invariance of System Functions
• Develop impulse invariance relation between system
functions
• Partial fraction expansion of transfer function
H s  

N
A k
c
• Corresponding impulse response

N
h t    
c k  1
k  1 s
 s k



A
e
s k t
0
t 
0
t  0
• Impulse response of discrete-time filter


 T dh c nT d 
 
h n
• System
function
N
s k nT d


N s k T d
n 

u n

u n   T d A k
e
k  1

k 
1
T d A k
e
 
H z
N T
A
d
k


k 
1
1  e
s k T d
z
 1
• Pole s=sk in s-domain transform into pole at e
s k T d
k
44

• Step 1: define specifications of filter
– Ripple in frequency bands
– Critical frequencies: passband edge, stopband edge, and/or cutoff frequencies.
– Filter band type: lowpass, highpass, bandpass, bandstop.
• Step 2: linear transform critical frequencies as follow
W = w/Td
• Step 3: select filter structure type and its order: Bessel, Butterworth,
Chebyshev type I, Chebyshev type II or inverse Chebyshev, elliptic.
• Step 4: convert Ha(s) to H(z) using linear transform in step 2.
• Step 5: verify the result. If it does not meet requirement, return to step 3.
Impulse Invariant Algorithm
45

Example: Impulse invariant method
Convert the analog filter with system function
Ha(s) = [s + 0.1]/[(s + 0.1)2 + 9]
into a digital IIR filter by means of the impulse invariance method.
The analog filter has a zero at s = - 0.1 and a pair of complex conjugate poles at pk = - 0.1 
j3. Thus,
H s  
a
1 1
2
 2
s  0 .1  j 3 s  0 .1  j
3
Then
H  z  
1
 2
1  e
46
1
2
 0 .1 T
e
j 3 T
z
 1
1  e
 0 .1 T
e
 j 3 T
z
 1

Frequency response
of digital filter.
Frequency response
of analog filter.
47

Disadvantage of previous
techniques: frequency
warping  aliasing effect
and error in specifications
48
of designed filter (frequencies)
So, prewarping of frequency
is considered.



H j 
Example
• Impulse invariance applied to
Butterworth 0 . 89125  H e j   1 0    0 . 2

0 . 3   

H e j   0.17783
• Since sampling rate Td cancels out we can assume Td=1
• Map spec to continuous time
0 . 89125  H  j   1
H  j   0.17783
0    0 . 2

0 . 3    

• Butterworth filter is monotonic so spec will be satisfied if
H c
j0 .2    0 .89125 and H c
j0 .3    0.17783 2

1
c
1  j / j
c 
2
N
• Determine N and c to satisfy these conditions
49

 1 
Example Cont’d
• Satisfy both constrains

2


1     

and 1     








 0 . 3   2 N 
1
 c 
0 .89125
  c

0 .17783
• Solve these equations to get
 0 . 2   2 N  1
 2
N  5 . 8858  6 and  c
 0 .
70474
sk  
• N must be an integer so we round it up to meet the spec
• Poles of transfer function
• 1 / 12 j    e
 j / 12 2 k  11 
c c
for k  0,1,...,11
The transfer function
H s   0 . 12093
 0 . 9945 s  0 . 4945 s 2
 1 . 3585 s  0 . 4945 
s 2
 0 . 364 s  0 . 4945
s 2
• Mapping to z-domain
H z   0 . 2871  0 . 4466 z  1  2 . 1428  1 . 1455 z  1

z  1
50
1  1 . 2971  0 . 6949 z  2 1  1 . 0691 z  1  0 . 3699 z  2
1 . 8557  0 . 6303 z  1

z  1
1  0 . 9972  0 . 257 z  2

Digital signal processing 22 scheme notes

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Digital signal processing 22 scheme notes