Digital signal processor part 3

Books.Books.
1.1. Digital Signal Processing Principles, Algorithms and ApplicationsDigital Signal Processing Principles, Algorithms and Applications
John G.Proakis & Dimitris G.Manolakis
2.2. Digital Signal ProcessingDigital Signal Processing By.
Sen M. Kuo & Woon-Seng Gan
3.3. Digital Signal Processing A Practical Approach.Digital Signal Processing A Practical Approach. By
Emmanuel C. Ifeachor & Barrie W. Jervis
4.4. Digital Signal Processing By Dr. Krishnanaik Vankdoth LAPDigital Signal Processing By Dr. Krishnanaik Vankdoth LAP
LAMBERT Academic Publishing Dnfscland/Germany – 2014LAMBERT Academic Publishing Dnfscland/Germany – 2014
krishnanaik.ece@gmail.com 2

3
Grading Policy
krishnanaik.ece@gmail.com
Assignments 1 & 2
20 Marks
Quiz Test -Mid-Term 30 Marks
Final Exam 50 Marks
Class will be divided different level as per their GPA
Group A- GPA
Group B- GPA
Group C – GPA

4
DTFT and DFT
• The DTFT of an aperiodic discrete time
signal is defined as
• The DFT of a signal is defined as
• Inverse DFT is defined as
∑
∞
−∞=
−
=
n
jwn
enxwX ][][
∑
−
=
π−
=
1N
0n
N/n2jk
e]n[x)k(X
∑
−
=
=
1
0
/2
][
1
][
N
k
Nknj
ekX
N
nx π
What is difference between DTFT and DFT?
(1)
(2)
(3)

5
• The DFT is periodic with period N.
Proof: ∑
−
=
π−
=
1N
0n
N/n2jk
e]n[x]k[X
∑
−
=
π+−
=+
1N
0n
N/n2)Nk(j
e]n[x]Nk[X
∑
−
=
π−π−
=
1N
0n
n2jN/n2jk
ee]n[x
Since e-j2πn
= 1
∑
−
=
π−
=+∴
1N
0n
N/n2jk
e]n[x]Nk[X
]k[X=
proved

6
Example 1: Find the DFT of the following sequence
[1 0 0 1]
∑∑∑ =
π−
=
π−
−
=
π−
===
3
0n
2/njk
3
0n
4/n2jk
1N
0n
N/n2jk
e]n[xe]n[xe]n[x]k[X
21001]3[x]2[x]1[x]0[x]n[x]0[X
3
0i
=+++=+++==∑=
2/3j
3
0n
2/njk
e]3[x00]0[xe]n[x]1[X π−
=
π−
+++== ∑
j1)sin(j)cos(1e.11 2
3
2
32/3j
+=−+=+= πππ−
( ) 0]3sin)3.[cos(11]3[]0[][]2[
3
0
3
=−+=+== ∑=
−−
ππππ
jexxenxX
n
jnj
∑=
− −
− = + = =
3
0
2/ 9 2/ 3
1 ]3[ ]0[ ] [ ]3[
n
j n j
j e x x e n x Xπ π

7
Example 2: Find the IDFT of the sequence
[2 1+j 0 1-j]
Solution:
Now
∑
−
=
π
=
1N
0k
N/n2jk
e]k[X
N
1
]n[x
[ ]]3[X]2[X]1[X]0[X
4
1
]k[X
4
1
]0[x
1N
0k
+++== ∑
−
=
∑∑ =
π
=
π
==
3
0k
2/jk
3
0k
4/2jk
e]k[X
4
1
e]k[X
4
1
]1[x
0e]3[Xe]2[Xe]1[X]0[X 2/3jj2/j
=+++= πππ
Similarly,
X[2] = 0 and X[3] = 1

8
Computational Complexity of the DFT
A large number of multiplications and
additions are required for the calculation
of the DFT.
Consider an 8-point DFT as given by
Let k2π/8 = K
∑=
π−
=
7
0n
8/n2jk
e]n[x]k[X
∑=
−
=
7
0n
jKn
e]n[x]n[x
7jK6jK5jK
4jK3jK2jK1jK0jK
e]7[xe]6[xe]5[x
e]4[xe]3[xe]2[xe]1[xe]0[x
−−−
−−−−−
++
+++++

9
There are eight complex multiplications and
seven complex additions. There are also
eight harmonic components to be evaluated.
Therefore, for an 8-point DFT:
Number of complex multiplications = 8×8
Number of complex additions = 8×7
For an N-point DFT
complex multiplications = N2
complex additions = N(N-1)
Clearly some means of reducing these is
required.

10
Decimation-in-time fast fourier transform
algorithm (Cooley-Tuckey Algorithm):
Notations: Equation (2) can be re-written as
Let
N/nk2j
1n
0N
n1 ex]k[X π−
−
=
∑= (4)
N/2j
N eW π−
= (5)
Also note that 2/
)2//(22)/2(2
][ N
NjNj
N WeeW === −− ππ
(6)
and
)2/N)(N/2(jk
N
2/N
N
k
N
)2/Nk(
N eWWWW π−+
==
( ) k
N
k
N
jk
N WsinjcosWeW −=π−π== π−
(7)

11
Summary:
N/2j
N eW π−
=
2/N
2
N WW =
k
N
)2/Nk(
N WW −=+
DFT: ∑
−
=
=
1N
0n
kn
Nn1 Wx]k[X (8)

12
Consider n data samples as:
x0x1x2x3………xn
Divide these samples into an even
numbered and odd numbered sequenes x2n
and x2n+1 respectively.
That is,
x2n = x0x2x4…..,xN-2
x2n+1 = x1x3x5….xN-1
Both of the above sequences contain N/2
points. krishnanaik.ece@gmail.com

13
Now equation (8) can be re-written as
follows:
k)1n2(
N
12/N
0n
1n2
nk2
N
12/N
0n
n21 WxWx]k[X +
−
=
+
−
=
∑∑ +=
∑∑
−
=
+
−
=
+=
12/N
0n
nk2
N1n2
12/N
0n
k
N
nk2
n2 WxWWx N
since
nk
2/N
nk2
n WW =
Therefore, ∑∑
−
=
+
−
=
+=
12/N
0n
nk
2/N1n2
k
N
12/N
0n
nk
2/Nn21 WxWWx]k[X
The above equation can be re-written as
]k[XW]k[X]k[X 12
k
N111 += (9)

14
Considering line 6 of the table it is seen that
4
k
4/N021 xwx]k[X += k = 0,1
Thus
4021 xx]0[X +=
while
404
2/2j
044/N021 xxxexxWx]1[X −=+=+= π−
similarly
7324
5123
6222
xx]0[X
xx]0[X
xx]0[X
+=
+=
+=
7324
5123
6222
xx]1[X
xx]1[X
xx]1[X
−=
−=
−=
We observe that the values with k = 1 differ only by a sign from
those with k = 0.

15
Now ]k[XW]k[X]k[X 22
k
2/N2111 += (10)
So, ]0[X]0[X]0[XW]0[X]0[X 222122
0
2/N2111 +=+= (11)
]1[jX]1[Xe]1[X]1[XW]1[X]1[X 2221
2/j
2122
1
2/N2111 −=+=+= π−
(12)
]2[X]2[X]2[Xe]2[X]2[XW]2[X]2[X 222122
22)8/2(j
2122
2
2/N2111 −=+=+= ×π−
Now ]0[XxxxWxxWx]2[X 21404
2
204
2
2/N021 =+=+=+=
and ]0[XxxxWx]2[X 22626
2
4/N222 =+=+=
(13)
Hence equation (13) is equivalent to
]0[X]0[X]2[X 222111 +=
]3[XW]3[X]3[X 22
3
2/N2111 +=
(14)
(15)

16
Now
]1[XxxxexxexxWx]3[X 21404
3j
04
3)2/2(j
04
3
4/N021 =−=+=+=+= π−π−
and ]1[Xxx]3[X 226222 =−=
Hence equation (15) is equivalent to
]1[jX]1[X]1[Xe]1[X]3[X 222122
3)4/2(j
2111 +=+= π−
(16)
Drawing these results together gives
]1[XW]1[X]1[jX]1[X]3[X
]1[XW]1[X]1[jX]1[X]1[X
]0[XW]0[X]0[X]0[X]2[X
]0[XW]0[X]0[X]0[X]0[X
22
2
821222111
22
2
821222111
22
0
821222111
22
0
821222111
−=+=
+=−=
−=−=
+=+=
(17)
The above equations are known as recomposition equations.

17
The number of complex additions and
multiplications involved is reduced in this way
because:
(i) the recomposition equations are expressed in
terms of powers of the recurring factor WN.
(ii) use is also made of relationships of the type
X21[2] = X21[0] and X21[3] = X 21[1] and
(iii) the presence of only sign differences in the
pairs of expressions is exploited.
The algorithm is known as the Cooley-Tukey
algorithm.
It can be shown that
Number of complex multiplications = (N/2)log2N

Frequency Domain Vs. Time Domain
18krishnanaik.ece@gmail.com

Fourier Transform
• A fourier transform is an useful analytical
tool that is important for many fields of
application in the digital signal processing.
• In describing the properties of the fourier
transform and inverse fourier transform, it
is quite convenient to use the concept of
time and frequency.
• In image processing applications it plays a
critical role. 19krishnanaik.ece@gmail.com

Fast Fourier transform
• Fast fourier transform proposed by Cooley
and Tukey in 1965.
• The fast fourier transform is a highly efficient
procedure for computing the DFT of a finite
series and requires less number of
computations than that of direct evaluation of
DFT.
• The FFT is based on decomposition and
breaking the transform into smaller transforms
and combining them to get the total transform.

FFT

DiscrDiscrete Fourier Transformete Fourier Transform
The DFT pair was given as
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
[ ] ( )
∑
−
=
π
=
1N
0k
knN/2j
ekX
N
1
]n[x
[ ] ( )
∑
−
=
−
=
1
0
/2
][
N
n
knNj
enxkX π

What is FFT?
• The fast fourier is an algorithm used to
compute the DFT. It makes use of the
symmetry and periodicity properties of twiddle
factor wN to effectively reduce the DFT
computation time.
• It is based on the fundamental principle of
decomposing the computation of DFT of a
sequence of length N into successively smaller
DFT.

Symmetry and periodicity
Symmetry
Periodicit
y
)()(
)()(
)(
kNn
N
nNk
N
kn
N
nNk
N
Nnk
N
kn
N
kn
N
kn
N
WWW
WWW
WW
−−−
++
−∗
==
==
=
k
N
N/k
N
N/
N
mnk
mN
nk
N
mnk
mN
nk
N
WWW
WWWW
−=−=
==
+ )2(2
/
/
,1
,

• FFT algorithm provides speed increase factors,
when compared with direct computation of the
DFT, of approximately 64 and 205 for 256
point and 1024 point transforms respectively.
• The number of multiplications and additions
required to compute N-point DFT using radix-
2 FFT are Nlog2N and N/2 log2N respectively.

• Example:
The number of complex multiplications required
using direct computation is
N2
=642
=4096
The number of complex multiplications required
using FFT is
N/2log2N=64/2log264=192
Speed improvement factor =4096/192= 21.33.

FFT Algorithms
• There are basically two types of FFT
algorithms.
• They are:
1. Decimation in Time
2. Decimation in frequency

Decimation in time
• DIT algorithm is used to calculate the DFT of
a N-point sequence.
• The idea is to break the N-point sequence
into two sequences, the DFTs of which can
be obtained to give the DFT of the original
N-point sequence.
• Initially the N-point sequence is divided into
N/2-point sequences xe(n) and x0(n) ,
which have even and odd numbers of x(n)
respectively.

• The N/2-point DFTs of these two sequences
are evaluated and combined to give the N-
point DFT.
• Similarly the N/2-point DFTs can be expressed
as a combination of N/4-point DFTs.
• This process is continued until we are left with
two point DFT.
• This algorithm is called decimation-in-time
because the sequence x(n) is often split into
smaller sequences.

Radix-2 DIT- FFT Algorithm
Radix-2: the sequence length N
satisfied:
L is an integer
L
N 2=
 To decompose an N point time domain
signal into N signals each containing a
single point. Each decomposing stage
uses an interlace decomposition,
separating the even- and odd-indexed
samples;
 To calculate the N frequency spectra
corresponding to these N time domain
signals.

0 4 8 12 2 6 10 14 1 5 9 13 3 7 11 15
0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15
0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 signal of
16 points
0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 152 signals
of 8 points
4 signals
of 4 points
8 signals
of 2 points
16 signals
of 1 point 31krishnanaik.ece@gmail.com

 Algorithm principle
 To divide N-point sequence x(n) into two N/2-
point sequence x1(r) and x2(r)
1
2
,2,1,0,)12()();2()( 21 −=+==
N
rrxrxrxrx 
 To compute the DFT of x1(r) and x2(r)
)1
2
~0()12()()(
)1
2
~0()2()()(
1
2
0 2
1
2
0 2
22
1
2
0 2
1
2
0 2
11
−=+==
−===
∑∑
∑∑
−
=
−
=
−
=
−
=
N
kWrxWrxkX
N
kWrxWrxkX
N
r
rk
N
N
r
rk
N
N
r
rk
N
N
r
rk
N

 To compute the DFT of N-point sequence x(n)
)1,2,1,0()()(
)()(
)12()2(
)()()()(
21
1
2
0 2
2
1
2
0 2
1
1
2
0
)12(
1
2
0
2
1
)(0
1
)(0
1
0
−=+=
+=
++=
+==
∑∑
∑∑
∑∑∑
−
=
−
=
−
=
+
−
=
−
=
−
=
−
=
NkkXWkX
WrxWWrx
WrxWrx
WnxWnxWnxkX
k
N
N
r
rk
N
k
N
N
r
rk
N
N
r
kr
N
N
r
rk
N
N
oddn
nk
N
N
evenn
nk
N
N
n
nk
N


)1
2
,1,0()()(
)
2
()
2
()
2
(
)1
2
,1,0()()()(
21
2
)
2
(
1
21
−=−=
+++=+
−=+=
+
N
kkXWkX
N
kXW
N
kX
N
kX
N
kkXWkXkX
k
N
N
k
N
k
N


)(nx
)(1 rx
)(2 rx
)(1 kX
)(2 kX
)(kX

 Butterfly computation flow graph
)1
2
,1,0()()()
2
(
)1
2
,1,0()()()(
21
21
−=−=+
−=+=
N
kkXWkX
N
kX
N
kkXWkXkX
k
N
k
N


)(1 kX
)(2 kX
k
NW
)()( 21 kXWkX k
N+
)()( 21 kXWkX k
N−
1−
There are 1 complex multiplication and 2 complex additions

N/2-
point
DFT
N/2-
point
DFT
)0(1X
)1(1X
)2(1X
)3(1X
)0(2X
)1(2X
)2(2X
)3(2X
0
NW
1
NW
2
NW
3
NW
)0()0(1 xx =
)2()1(1 xx =
)4()2(1 xx =
)6()3(1 xx =
)1()0(2 xx =
)3()1(2 xx =
)5()2(2 xx =
)7()3(2 xx =
)(1 rx
)(2 rx
)4(X1−
)5(X1−
)6(X1−
)7(X1−
)0(X
)1(X
)2(X
)3(X
N-point DFT 36krishnanaik.ece@gmail.com

fo
r
3
2=N
)(nx
2-point
DFT
2-point
DFT
2-point
DFT
2-point
DFT
Synthesize
the 2-point
DFTs into a
4-point DFT
Synthesize
the 2-point
DFTs into a
4-point DFT
Synthesize
the 4-point
DFTs into a
8-point DFT
)(kX
3-stage synthesize, each has N/2 butterfly
computation
 The computation complexity

•At the end of computation flow graph at any
stage, output variables can be stored in the
same registers previously occupied by the
corresponding input variables.
•This type of memory location sharing is
called in-place computation which results in
significant saving in overall memory
requirements.

 The distance between two nodes in a butterfly
For there are L stagesL
N 2=
StageStage DistanceDistance
stage 1stage 1 11
stage 2stage 2 22
stage 3stage 3 44
stagestage LL

1
2 −L

 Bit-reversed order
In the DFT computation scheme, the DFT samples X(k)
appear at the output in a sequential order while the input
samples x(n) appear in a different order: a bit-reversed
order.
Thus, a sequentially ordered input x(n) must be
reordered appropriately before the fast algorithm can be
implemented.
Let m, n represent the sequential and bit-reversed order
in binary forms respectively, then:
m: 000 001 010 011 100 101 110 111
n: 000 100 010 110 001 101 011 111

 Why is the input bit-reversed order
1n 2n
)( 012 nnnx
0n
0
1
0
1
0
1
0
1
0
1
0
1
0
1
)000(x
)100(x
)010(x
)110(x
)001(x
)101(x
)011(x
)111(x
)(0x
)(4x
)(2x
)(6x
)(1x
)(5x
)(3x
)(7x

 How to get the bit-reversed order
Let represent the natural order, the represent
the bit-reversed order, then:
n nˆ
)ˆ()(ˆ nxnxnn ⇔> ,if
)0(A )1(A )2(A )3(A )4(A )5(A )6(A )7(A
)0(x )1(x )2(x )3(x )4(x )5(x )6(x )7(xn
nˆ )0(x )7(x)1(x)4(x )6(x)2(x )3(x)5(x

Decimation-In-Frequency
• It is a popular form of FFT algorithm.
• In this the output sequence x(k) is divided into
smaller and smaller subsequences, that is why
the name decimation in frequency,
• Initially the input sequence x(n) is divided into
two sequences x1(n) and x2(n) consisting of
the first n/2 samples of x(n) and the last n/2
samples of x(n) respectively

Radix-2 DIF- FFT Algorithm
 Algorithm principle
 To divide N-point sequence x(n) into two N/2-
point sequence
1
2
0),
2
(
1
2
0),(
−≤≤+
−≤≤
N
n
N
nx
N
nnxThe former N/2-point
The latter N/2-point

0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
0 1 2 3 0 1 2 3
butterfly computation
0 1 2 3 0 1 2 3
butterfly computation butterfly computation
0 1 0 1 0 1 0 1
butterfly butterfly butterfly butterfly
0 4 2 6 1 5 3 7
0 1 0 1 0 10 1
)(nx
)(kX 45krishnanaik.ece@gmail.com

 To compute the DFT of N-point sequence x(n)
)1,1,0()
2
()1()(
)
2
()(
)
2
()(
)()()()(
1
2
0
1
2
0
2
1
2
0
)
2
(
1
2
0
1
2
1
2
0
1
0
−=


 +−+=






++=
++=
+==
∑
∑
∑∑
∑∑∑
−
=
−
=
−
=
+
−
=
−
=
−
=
−
=
NkW
N
nxnx
W
N
nxWnx
W
N
nxWnx
WnxWnxWnxkX
N
n
nk
N
k
N
n
nk
N
k
N
N
N
n
k
N
n
N
N
n
nk
N
N
N
n
nk
N
N
n
nk
N
N
n
nk
N


 To separate the even and odd numbered
samples of X(k)
)1
2
,,1,0(,12,2let −=+==
N
rrkrk 
)1
2
,1,0()
2
()()2(
1
2
0 2
−=


 ++= ∑
−
=
N
rW
N
nxnxrX
N
n
nr
N 
)1
2
,1,0()
2
()()12(
1
2
0 2
−=


 +−=+ ∑
−
=
N
rWW
N
nxnxrX
N
n
nr
N
n
N 

)1
2
,1,0()()12(
)1
2
,1,0()()2(
1
2
0 2
2
1
2
0 2
1
−==+
−==
∑
∑
−
=
−
=
N
rWnxrX
N
rWnxrX
N
n
nr
N
N
n
nr
N


1
2
,1,0
)
2
()()(
)
2
()()(
let
2
1
−=












+−=
++=
N
n
W
N
nxnxnx
N
nxnxnx
n
N


)(nx
)
2
(
N
nx +
n
NW
1−
)
2
()()(1
N
nxnxnx ++=
n
NW
N
nxnxnx )
2
()()(2 





+−=
 Butterfly computation flow graph
There are 1 complex multiplication and 2 complex additions

fo
r
3
2=N
N/2-
point
DFT
N/2-
point
DFT
)0(X
)2(X
)4(X
)6(X
)1(X
)3(X
)5(X
)7(X
)0(1x
)1(1x
)2(1x
)3(1x
)0(x
)1(x
)2(x
)3(x
)4(x
)5(x
)6(x
)7(x
)3(2x
3
NW
1−
)2(2x
2
NW
1−
)1(2x
1
NW
1−
)0(2x0
NW
1−

fo
r
3
2=N
)0(x
)2(x
)1(x
)3(x
)4(x
)6(x
)5(x
)7(x
)0(X
)4(X
)2(X
)6(X
)1(X
)5(X
)3(X
)7(X
1−
1−
1−
1−
0
NW
1
NW
2
NW
3
NW
1−
1−
1−
1−
1−
0
NW
1−
0
NW
0
NW
2
NW
1−
0
NW
1−
0
NW
0
NW
2
NW

 The comparison of DIT and DIF
 The order of samples
DIT-FFT: the input is bit- reversed order and the
output is natural order
DIF-FFT: the input is natural order and the output is
bit- reversed order
 The butterfly computation
DIT-FFT: multiplication is done before additions
DIF-FFT: multiplication is done after additions

 Both DIT-FFT and DIF-FFT have the identical
computation complexity. i.e. for , there
are total L stages and each has N/2 butterfly
computation. Each butterfly computation has 1
multiplication and 2 additions.
L
N 2=
 Both DIT-FFT and DIF-FFT have the
characteristic of in-place computation.
 A DIT-FFT flow graph can be transposed to a
DIF-FFT flow graph and vice versa.

Dr. V. Krishnanaik Ph.D
FIR and IIR Filter Design
Techniques
56

57
Outline
• Introduction
• IIR Filter Design by Impulse invariance
method
• IIR Filter Design by Bilinear transformation
method
• FIR Filter Design by Window function
technique
• FIR Filter Design by Frequency sampling
technique
• FIR Filter Design by MSEkrishnanaik.ece@gmail.com

Introduction
• Basic filter classification
• We put emphasis on the digital filter now,
and will introduce to the design method of
the FIR filter and IIR filter respectively.
Filter
Analog Filter
Digital Filter
IIR Filter
FIR Filter

59
• IIR is the infinite impulse response abbreviation.
• Digital filters by the accumulator, the multiplier,
and it constitutes IIR filter the way, generally
may divide into three kinds, respectively is
Direct form, Cascade form, and Parallel form.
• IIR filter design methods include the impulse
invariance, bilinear transformation, and step
invariance.
• We must emphasize at impulse invariance and
bilinear transformation.

Continuous frequency
band transformation
Impulse Invariance
method
Bilinear
transformation
method
Step invariance
method
IIR filter
Normalized analog
lowpass filter
IIR filter design methods

61
Introduction
• The structures of IIR filter
Direct
form 1
Direct form2
b0
b1
b2 b2
b1
b0
-a1
-a2
-a1
-a2
x(n) x(n)Y(n) Y(n)
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−

62
Introduction
• The structures of IIR filter
Cascade form
x(n) Y(n)
b0
b1
b2
-a1
-a2
-c1
-c2
d1
d2
Parallel form
Y(n)x(n)
b1
b0
d1
d0
E
-c1
-c2
-a1
-a2
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−

63
Introduction
• FIR is the finite impulse response
abbreviation, because its design
construction has not returned to the part
which gives.
• Its construction generally uses Direct form
and Cascade form.

64
Introduction
• FIR filter design methods include the window
function, frequency sampling, minimize the
maximal error, and MSE.
• We must emphasize at window function,
frequency sampling, and MSE.
Window
function
technique
Frequency
sampling
technique
Minimize the
maximal error
FIR filter
Mean
square
error

65
Introduction
• The structures of FIR filter
x(n) x(n)
b1
b2
b3
b4
b0
Y(n) Y(n)
Direct form Cascade form
b1
b2
d1
d2
b0
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−

66
IIR Filter Design by Impulse invariance
method
• The most straightforward of these is the
impulse invariance transformation
• Let be the impulse response
corresponding to , and define the
continuous to discrete time transformation by
setting
• We sample the continuous time impulse
response to produce the discrete time filter
( )ch t
( )cH s
( ) ( )ch n h nT=

67
method
• The frequency response is the Fourier
transform of the continuous time function
and hence
'( )H ω
*
( ) ( ) ( )c c
n
h t h nT t nTδ
∞
=−∞
= −∑
1 2
'( ) ( )c
k
H H j k
T T
π
ω ω
∞
=−∞
 
= − 
 
∑

68
method
• The system function is
• It is the many-to-one transformation from the
s plane to the z plane.
1 2
( ) | )sT cz e
k
H z H s jk
T T
π∞
=
=−∞
 
= − 
 
∑

69
method
• The impulse invariance transformation does
map the - axis and the left-half s plane
into the unit circle and its interior,
respectively
jω
Re(Z)
Im(Z)
1
S domain Z domain
sT
e
jω
σ

70
method
• is an aliased version of
• The stop-band characteristics are maintained adequately
in the discrete time frequency response only if the
aliased tails of are sufficiently small.
'( )H ω ( )cH jω
0 ω
'( )H ω
/Tπ 2 /Tπ
( )cH jω

71
method
• The Butterworth and Chebyshev-I lowpass
designs are more appropriate for impulse
invariant transformation than are the
Chebyshev-II and elliptic designs.
• This transformation cannot be applied
directly to highpass and bandstop designs.

72
method
• is expanded a partial fraction expansion
to produce
• We have assumed that there are no multiple
poles
• And thus
( )cH s
1
( )
N
k
c
k k
A
H s
s s=
=
−
∑
1
( ) ( )k
N
s t
c k
k
h t A e u t
=
= ∑
1
( ) ( )k
N
s nT
k
k
h n A e u n
=
= ∑
1
1
( )
1 k
N
k
s T
k
A
H z
e z−
=
=
−
∑

73
method
• Example:
Expanding in a partial fraction
expansion, it produce
The impulse invariant transformation
yields a discrete time design with the
system function
2 2
( )
( )
c
s a
H s
s a b
+
=
+ +
1/ 2 1/ 2
( )cH s
s a jb s a jb
= +
+ + + −
( ) 1 ( ) 1
1/ 2 1/ 2
( )
1 1a jb T a jb T
H z
e z e z− + − − − −
= +
− −

74
IIR Filter Design by Bilinear
transformation method
• The most generally useful is the
bilinear transformation.
• To avoid aliasing of the frequency response as
encountered with the impulse invariance
transformation.
• We need a one-to-one mapping from the s plane
to the z plane.
• The problem with the transformation is
many-to-one.
sT
z e=

75
• We could first use a one-to-one transformation
from to , which compresses the entire s
plane into the strip
• Then could be transformed to z by
with no effect from aliasing.
s 's
Im( ')s
T T
π π
− ≤ ≤
's
's T
z e=
σ
jω
'σ
jω
/Tπ−
/Tπ
s domain s’ domain

76
• The transformation from to is given by
• The characteristic of this transformation is
seen most readily from its effect on the
axis.
• Substituting and , we obtain
s 's
12
' tanh ( )
2
sT
s
T
−
=
jω
s jω= ' 's jω=
12
' tan ( )
2
T
T
ω
ω −
=

77
• The axis is compressed into the interval
for in a one-to-one method
• The relationship between and is
nonlinear, but it is approximately linear at
small .
( ,
T T
π π
−
'ω
ω
ω 'ω
'ω ω≈
-
ω
'ω
/Tπ
/Tπ−

78
• The desired transformation to is now
obtained by inverting to produce
• And setting , which yields
12
' tanh ( )
2
sT
s
T
−
=
2 '
tanh( )
2
s T
s
T
=
s z
1
' ( )lns z
T
=
2 ln
tanh( )
2
z
s
T
=
1
1
2 1
( )
1
z
T z
−
−
−
=
+
Re(Z)
Im(Z)
1
S domain Z domain
1
2
1
2
T
s
z
T
s
+
=
−
jω
σ

79
• The discrete-time filter design is obtained
from the continuous-time design by means of
the bilinear transformation
• Unlike the impulse invariant transformation,
the bilinear transformation is one-to-one, and
invertible.
1 1
(2/ )(1 )/(1 )
( ) ( ) |c s T z z
H z H s − −
= − +
=

80
FIR Filter Design by Window function
technique
• Simplest FIR the filter design is window
function technique
• A supposition ideal frequency response may
express
where
( ) [ ]j j n
d d
n
H e h n eω ω
∞
−
=−∞
= ∑
1
[ ] ( )
2
j j n
d dh n H e e d
π
ω ω
π
ω
π −
= ∫

81
technique
• To get this kind of systematic causal FIR to
be approximate, the most direct method
intercepts its ideal impulse response!
[ ] [ ] [ ]dh n w n h n= g
( ) ( ) ( )dH W Hω ω ω= ∗

82
technique
• Truncation of the Fourier series produces
the familiar Gibbs phenomenon
• It will be manifested in , especially if
is discontinuous.
( )H ω
( )dH ω

83
technique
• 1.Rectangular window
• 2.Triangular window (Bartett window)
1, 0
[ ]
0,
n M
w n
otherwise
≤ ≤
= 

2 , 0
2
2[ ] 2 ,
2
0,
n Mn
M
n Mw n n M
M
otherwise
 ≤ ≤


= − < ≤



84
technique
• 1.Rectangular window
• 2.Triangular window (Bartett window)
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Rectangular window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Bartlett window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
-50
0
50
100
pi unitsFrequencyresponseT(jw)(dB)
Rectangular window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
-50
0
50
100
pi units
FrequencyresponseT(jw)(dB)
Bartlett window

85
technique
• 3.HANN window
• 4.Hamming window
1 2
1 cos , 0
[ ] 2
0,
n
n M
w n M
otherwise
π  
− ≤ ≤  =  


2
0.54 0.46cos , 0
[ ]
0,
n
n M
w n M
otherwise
π
− ≤ ≤
= 


86
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
-50
0
50
100
pi units
Hanning window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
-50
0
50
100
pi units
Hamming window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Hanning window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Hamming window
technique
• 3.HANN window
• 4.Hamming window

87
technique
• 5.Kaiser’s window
• 6.Blackman window
2
0
0
2
[ 1 (1 ) ]
[ ] , 0,1,...,
[ ]
n
I
Mw n n M
I
β
β
− −
= =
2 4
0.42 0.5cos 0.08cos , 0
[ ]
0,
n n
n M
w n M M
otherwise
π π
− + ≤ ≤
= 


88
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-100
-50
0
50
100
pi units
Blackman window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-150
-100
-50
0
50
100
pi units
Kaiser window
• 5.Kaiser’s window
• 6.Blackman window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Blackman window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Kaiser window
technique

89
technique
( / )s Mω
Window Peak sidelobe level
(dB)
Transition
bandwidth
Max. stopband
ripple(dB)
Rectangular -13 0.9 -21
Hann -31 3.1 -44
Hamming -41 3.3 -53
Blackman -57 5.5 -74

90
FIR Filter Design by Frequency sampling
technique
• For arbitrary, non-classical specifications of
, the calculation
of , n=0,1,…,M, via an appropriate
approximation can be a substantial
computation task.
• It may be preferable to employ a design
technique that utilizes specified values of
directly, without the necessity of
determining
' ( )dH ω
( )dh n
' ( )dH ω ( )dh n

91
technique
• We wish to derive a linear phase IIR filter
with real nonzero . The impulse
response must be symmetric
where are real and denotes the
integer part
( )h n
[ /2]
0
1
2 ( 1/ 2)
( ) 2 cos( )
1
M
k
k
k n
h n A A
M
π
=
+
= +
+
∑
kA [ / 2]M
0,1,...,n M=

92
technique
• It can be rewritten as
where and
• Therefore, it may write
where
1
/ 2 /
0
/2
( )
N
j k N j kn N
k
k
k N
h n A e eπ π
−
=
≠
= ∑ 0,1,..., 1n N= −
1N M= + k N kA A −=
/ 2 /
( ) j k N j kn N
k kh n A e eπ π
=
1
0
/2
( ) ( )
N
k
k
k N
h n h n
−
=
≠
= ∑
0,1,..., 1n N= −

93
technique
• with corresponding transform
where
• Hence
which has a linear phase
1
0
/2
( ) ( )
N
k
k
k N
H z H z
−
=
≠
= ∑
/
2 / 1
(1 )
( )
1
j k N N
k
k j k N
A e z
H z
e z
π
π
−
−
−
=
−
' ( 1)/2 sin / 2
( )
sin[( / / 2)]
j T N
k k
TN
H A e
k N T
ω ω
ω
π ω
− −
=
−

94
technique
• The magnitude response
which has a maximum value
at where
' sin / 2
( )
sin[( / / 2)]
k k
TN
H A
k N T
ω
ω
π ω
=
−
kN A
/k sk Nω ω= 2 /s Tω π=

95
technique
• The only nonzero contribution to at
is from , and hence that
• Therefore, by specifying the DFT samples of
the desired magnitude
response at the frequencies , and
setting
'( )H ω
kω ω= '
( )kH ω
'( )k kH N Aω =
'
( )dH ω kω
'
( ) /k d kA H Nω= ±

96
technique
• We produce a filter design from equation (5.1)
for which
• The desired and actual magnitude responses are
equal at the N frequencies
'
'( ) ( )k d kH Hω ω=
kω

97
technique
• In between these frequencies, is
interpolated as the sum of the responses
, and its magnitude does not, equal that of
'( )H ω
'
( )kH ω
'
( )dH ω

98
technique
• Example: For an ideal lowpass filter
from , we would choose
• The frequency samples are indeed equal
to the desired
' 1, 0,1,...,
( )
0, 1,...,[ / 2]
d k
k P
H
k P M
ω
=
= 
= +
'
( ) /k d kA H Nω= ±
( 1) / ( 1), 0,1,...,
0, 1,...,[ / 2]
k
k
M k P
A
k P M
 − + =
= 
= +
'
( )kH ω
'
( )d kH ω

99
technique
• The response is very similar to the result
form using the rectangular window, and the
stopband is similarly disappointing.
• We can try to search for the optimum value
of the transition sample would quickly lead
us to a value of approximately
, k p≠0.38( 1) /( 1)p
pA M= − +

100
FIR Filter Design by MSE
• : The spectrum of the filter we obtain
• : The spectrum of the desired filter
• MSE =
( )H f
( )dH f
( ) ( )∫−
−
−
2/
2/
21 s
s
f
f ds dffHfHf
0 0.1 0.2 0.3 0.4 0.5
-0.5
0
0.5
1
1.5

101
• Larger MSE, but smaller maximal error
•
• Smaller MSE, but larger maximal error
0 0.1 0.2 0.3 0.4
-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4
-0.5
0
0.5
H(F) H(F) - H (F)d
0 0.1 0.2 0.3 0.4
-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4
-0.5
0
0.5
H(F) H(F) - H (F)d

102
• 1.
( ) ( ) ( ) ( )∫∫ −−
−
−=−=
2/1
2/1
22/
2/
21
dFFHFRdffHfRfMSE d
f
f ds
s
s
( ) ( ) dFFHFnns d
k
n
∫ ∑−
=
−=
2/1
2/1
2
0
|| 2cos][ π
( ) ( ) ( ) ( ) dFFHFnnsFHFnns d
k
n
d
k
n
∫ ∑∑−
==






−





−=
2/1
2/1
00
2cos][2cos][ ππ
( ) ( )
1/2
1/2
0 0
[ ]cos 2 [ ]cos 2
k k
n
s n n F s F dF
τ
π τ π τ
−
= =
= ∑ ∑∫
( ) ( ) ( )
1/2 1/2
2
1/2 1/2
0
2 [ ]cos 2
k
d d
n
s n n F H F dF H F dFπ
− −
=
− +∑∫ ∫

103
• 2. when n ≠ τ,
when n = τ, n ≠ 0,
when n = τ, n = 0,
• 3. The formula can be repressed as:
( ) ( ) 02cos2cos
2/1
2/1
=∫−
dFFFn τππ
( ) ( ) 2/12cos2cos
2/1
2/1
=∫−
dFFFn τππ
( ) ( ) 12cos2cos
2/1
2/1
=∫−
dFFFn τππ
( ) ( ) ( )dFFHdFFHFnnsnssMSE dd
k
n
k
n
∫∫ ∑∑ −−
==
+−+=
2/1
2/1
22/1
2/1
01
22
2cos][22/][]0[ π

104
• 4. Doing the partial differentiation:
• 5. Minimize MSE: for all n’s
( )∫−
−=
∂
∂ 2/1
2/1
2]0[2
]0[
dFFHs
s
MSE
d ( ) ( )∫−
−=
∂
∂ 2/1
2/1
2cos2][
][
dFFHFnns
ns
MSE
dπ
0
][
=
∂
∂
ns
MSE
( )∫−
=
2/1
2/1
]0[ dFFHs d ( ) ( )∫−
=
2/1
2/1
2cos2][ dFFHFnns dπ
[ ] [0]
[ ] [ ]/ 2 for n=1,2,...,k
[ ] [ ]/ 2 for n=1,2,...,k
[ ] 0 for n<0 and n N
h k s
h k n s n
h k n s n
h n
=
+ =
− =
= ≥

105
• IIR advantage:
1. It is easy to design
2. It is easy to implementation
• IIR disadvantage:
1. Infinite impulse response
2. It is hard to optimalize than FIR
3. Non-stable

Dr. Krishnanaik VankdothDr. Krishnanaik Vankdoth
B.EB.E(ECE),(ECE), M.TechM.Tech (ECE),(ECE), Ph.DPh.D (ECE)(ECE)
Professor in ECE Dept
Aksum University, Ethiopia– 1010
Krishnanaik.ece@gmail.com
Krishnanaik_ece@yahoo.com
Phone : +919441629162

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