Lecture 4: Frequency domain
representation, DTFT, IDTFT, DFT, IDFT
Instructor:
Dr. Gleb V. Tcheslavski
Contact: gleb@ee.lamar.edu
Office Hours: Room 2030
Class web site:
http://ee.lamar.edu/gleb/dsp/ind
ex.htm
ELEN 5346/4304
DSP and Filter Design
Fall 2008
1
2
Some history
Jean Baptiste Joseph Fourier was born
in France in 1768. He attended the Ecole Royale
Militaire and in 1790 became a teacher there.
Fourier continued his studies at the Ecole
Normale in Paris, having as his teachers
Lagrange, Laplace, and Monge. Later on, he,
together with Monge and Malus, joined Napoleon
as scientific advisors to his expedition to Egypt
where Fourier established the Cairo Institute.
In 1822 Fourier has published his most
famous work: The Analytical Theory of Heat.
Fourier showed how the conduction of heat in
solid bodies may be analyzed in terms of infinite
mathematical series now called by his name, the
Fourier series.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
3
Frequency domain representation
Letxn  Acos(0n   )a (co)sinusoidal input
(4.3.1)
frequency
A j (0n ) A  j (0n ) A j j0n A  j  j0n
xn  e
 e
 e e  e e
2
2
2
2
complex exponent of 0
complex exponent of -0
A sinusoidal signal is represented by TWO complex exponents of
opposite frequencies in the frequency domain.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.3.2)
4
Frequency domain representation (cont)


Letxn  e j0n  yn   hk e j0 ( nk )    hk e j0k  e j0n
k
 k



H e j0 xn
The frequencyresponse :H  e j    hk e jk  H  e j  e
(4.4.1)
 
j H e j
phase
(4.4.2)
k
magnitude
Iff it exists!
Forarealsystem :H  e j    hn e  j n   hn*e  j n
n
 H  e
n
 
 H e  e
e
 H e   H e 


 H  e    H  e 
j H e j
j
j
j
 j
DSP and Filter Design

 j H e j
(4.4.3)

 j
 j
H  e j  has a period of 2
ELEN 5346/4304
*


   hn e  jn   H *  e  j 
 n

Fall 2008
(4.4.4)
5
Frequency domain representation (cont 2)
10
For an arbitrary
real LTI system:
Magnitude (dB)
5
0
-5
-10
Symmetric with
respect to 
-15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized Frequency ( rad/sample)
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized Frequency ( rad/sample)
1.6
1.8
2
Anti-symmetric
with respect to 
Phase (degrees)
100
50
0
-50
-100
ELEN 5346/4304
DSP and Filter Design
Fall 2008
6
Frequency domain representation (cont 3)
Combining (4.3.2) and (4.4.4) – back to our sinusoid!




A j
A
e H e j0 e j0n  e  j H e  j0 e  j0n  realsystem
2
2
j H  e j0  j n
 j H  e j0   j n
A j
A  j
j0
j0
0
 e H e
e
e  e H e
e
e 0
2
2
j 0 n   H  e j0  
 j 0 n   H  e j0   
A
j0 




 H e
e

e


2


Stady-state response: yn 


LTI filtering:








yn  A H e j0 cos 0 n    H e j0

due to the
same as from the
input
the input input
change due to the system
phase change due to the system
Via design, we manipulate H(ej), therefore, hn, and, finally, manipulate the
coefficients in the Linear Constant Coefficient Difference Equation (LCCDE)
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.6.1)
(4.6.2)
(4.6.3)
(4.6.4)
7
Frequency domain representation (cont 4)
xn  e j0n ,n  0

 j n
n
 ytot ,n   ci i  ke 0
initialconditions 0
i
LCCDE:
ytot ,n  ke j0n
for large enough n:
yn   hk e
j0  n  k 
un  k
k
 n
 j0 k  j0 n
j0
j0 n
   hk e
e

H
(
e
)
e

 k 0

(4.7.1)
(4.7.2)
(4.7.3)
n 
H (e j )   hne jn   hn  if thesystemisBIBO,thefrequencyresponseexists
n
ELEN 5346/4304
DSP and Filter Design
n
Fall 2008
(4.7.4)
8
Frequency domain representation (cont 5)
for an LTI:
xn  e j0nun  yn  ytr ,n  yss,n
(4.8.1)
We don’t need systems of order higher than 2: can always make cascades.
yss,n  H (e j0 )e j0n existsonly forLTI ,BIBOsystems
for a real, LTI, BIBO system:
xn  Acos(0 n   ) yss ,n  A H (e j0 ) cos(0 n    H (e j0 )
effects of filtering
We cannot observe ANY frequency components in the output that are not
j
present in the input (in steady state). We may see less when H (e r )  0
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.8.2)
9
Frequency domain representation (cont 6)
In continuous time:
xt  st  nt {wewant} yt  c  st  distortion  lesstransmission
signal
noise

Y () 

const
yt e jt dt 

(4.9.1)
delay

 j ( t  )
 j
 j
c

s
e
d
(
t


)
e

ce
S ()  H ()S ()
t




We need a constant magnitude and linear phase for the frequencies of interest.
Ideal filters:
LPF
H (e
j0
HPF
H (e
)

j0
)

H (e
)
H (e
j0
BSF

Fall 2008
H (e j0 )
)

Ideal filters are non-realizable!
DSP and Filter Design
BPF


ELEN 5346/4304
j0




(4.9.2)
10
CTFT and ICTFT

X () 
CTFT:

x(t )e  jt dt
(4.10.1)
X ()e jt d 
(4.10.2)

ICTFT:
1
x(t ) 
2



Examples:
ELEN 5346/4304
DSP and Filter Design
Fall 2008
11
DTFT
X e
j

  x e
n 
 j n
n
if exists
acomplex function,  periodic,continuouson
(4.11.1)

(4.11.2)

X re  e j   X  e j  cos X  e j  , X im  e j   X  e j  sin
 X e  
j
X  e j   X re 2  e j   X im 2  e j 
2
(4.11.3)
What’s about convergence???
1. Absolute convergence:
if xn  l1i.e. xn   X  e j convergesabsolutelyoruniformly
n
j
Let X k (e ) 
ELEN 5346/4304
k

n  k
DSP and Filter Design
xn e jn lim X  e j   X k  e j   0
k 
abs .error
Fall 2008
(4.11.4)
(4.11.5)
12
DTFT (cont)
Absolutely summable sequences always have finite energy. However,
finite energy sequences are not necessary absolutely summable.


2
an
are

not

l
.

In

fact
,

they

are

not

l
:
x


1
2  n
n
Acos(0 n   ) 

must be
un
2. Mean-square convergence:

lim
k 

X  e j   X k  e j   0
(4.12.1)

energyof theerror
The total energy of the error must approach zero, not an error itself!
ELEN 5346/4304
DSP and Filter Design
Fall 2008
13
IDTFT
Let p  2 isa period , X  e j  

xe
Combining (4.11.1) and (4.12.2)
 1
xl 

l 
 2

X  e j 
ELEN 5346/4304

 e
j ( n l )

2
n
p
n
n 
1
1
j
j n
x

X
(
e
)
e
d


IDTFT: n
p p
2
j


X (e j )e jn d
(4.13.2)

1
xn 
2

 
 jl  jn
x
e
l
 e d
  l


 
sin  (n  l ) 
d    xl
  xl nl  xn
 (n  l )

 l 
shows where xn “lives” in the frequency domain.
DSP and Filter Design
(4.13.1)
Fall 2008
(4.13.3)
(4.13.4)
14
Back to ideal filters
1
Ideal LPF:
H (e j0 )
0

c 
2
H LP  e
j


1,0    c


0,c    
(4.14.1)
Using IDTFT:
hLP ,n
1

2
c
 e

j n
c
1  e jc n e  jc n
d 


2  jn
jn
 si n  c n
,   n  

n

(4.14.2)
1. The response in (4.14.2) is not absolutely summable, therefore, the filter is
not BIBO stable!
2. The response in (4.14.2) is not causal and is of an infinite length.
As a result, the filter in (4.14.1) is not realizable.
Similar derivations show that none of the ideal filters in slide 9 is realizable.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
15
DTFT properties
Linearity :ag n  bxn   aG  e j   bX  e j 
(4.15.1)
Shiftintime :g n  n0   e  j n0 G  e j 
(4.15.2)
DTFT
DTFT


dG  e 
Differentiation :n  g    j
Shiftin frequency :e  j0 n g n   G e j ( 0 ) 
(4.15.3)
DTFT
j
d
Linearconvolution :g n  xn   G  e j   X  e j 
n
(4.15.4)
DTFT
(4.15.5)
DTFT
1
Periodicconvolution :g n  wn   
DTFT 2

Parseval ' stheorem :  g n  w*n 
n 
ELEN 5346/4304
DSP and Filter Design

j
j (  )
G
e
W
e



 d

(4.15.6)

1
2
 continuous, periodic functions
j
*
j
 G  e W  e  d

Fall 2008
(4.15.6)
16
DTFTs of commonly used sequences
ELEN 5346/4304
DSP and Filter Design
Fall 2008
17
DTFT examples
Letxn  e j0n    X  e j   2  (  0  2 k )
(4.17.1)
0  xn  1   X  e j   2  (  2 k )
(4.17.2)
DTFT
k
DTFT
k
un  un 1   n   U  e j   1  e j   1 U  e j  
DTFT
1
 c 2   (  l 2 )
 j
1 e
l
(4.17.3)
½ of DC value of un
1
un   
    (  l 2 )
DTFT 1  e  j
l
(4.17.4)
*

*  j n
j n 
*
j
h
e

h
e

H
e


n n

n


 n


1
1
 j n
j
j n
 j n
h
g
e

H
e
e
g
e
d




n n n
n 2 
n
2

1

2
ELEN 5346/4304
DSP and Filter Design


 H  e  G  e
j
j (   )
 d

Fall 2008
(4.17.5)


 H  e   g e
j
n

 j (  ) n
d
n
(4.17.6)
18
DTFT examples (cont)
We can re-work the Parseval’s theorem (4.15.6) as follows:
Energy En   hn
2
n
1

2

1
H
e
H
e
d


    
2
j
*
j

 H  e 

j
2
d
(4.18.1)
Sh ( e j )
energy density (spectrum)
Autocorrelation function:
rgg ,l   gn gnl *   gn g(l n )*  gl  gl *  G  e j  G*  e j   Sg  e j 
n
ELEN 5346/4304
DSP and Filter Design
n
DTFT
Fall 2008
(4.18.2)
19
DTFT examples (cont 2)
One obvious problem with DTFT is that we can never compute it since xn
needs to be known everywhere! which is impossible!
Therefore, DTFT is not practical to compute.
Often, a finite dimension LTI system is described by LCCDE:
N
M
N
 ai yni   bm xnm    ai e
i 0
DTFT
m 0
i 0
M
Y e   b e


j
X e
j
m0
N
i 0
j
  b e
M
m0
m
 j m
m
 a e
Y e
 ji
 ji
 H  e j 
i
Prediction of steady-state behavior of LCCDE
ELEN 5346/4304
DSP and Filter Design
Fall 2008
 j m
X  e j 
(4.19.1)
practical (finite
dimensions)
(4.19.2)
How to measure frequency response
of an actual (unknown) filter?
20
1. Perform two I/O experiments:


 
 
 
 
1)xn ( c )  Acos(0 n   ) yn ( c )  A H e j0 cos 0 n    H e j0  ytr ,n ( c )

LTI

j
j
(s)
(s)
(s)
2)xn  Asin(0 n   ) yn  A H e 0 sin 0 n    H e 0  ytr ,n

LTI
(4.20.1)
(4.20.2)
2. Analyze these measurements and form:
 xn  xn ( c )  jxn ( s )  Ae j (0 n  )


j 0 n  H  e j0 
j0
(c)
(s)

e
 ytr ,n
 yn  yn  jyn  A H e
jH  e j0 
yn
j0
Finally  H e
e
 ztr ,n
xn




That’s a good way to measure/estimate a frequency response for every .
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.20.3)
(4.20.4)
(4.20.5)
21
DFT and IDFT
Consider an N-sequence xn (at most N non-zero values for 0  n  N-1)
X e
j
  x e
 j n
n
  xn e jn  X  e j 
n 0
n
N 1
DFT:
N 1
X k   xn e
2
 j kn
N
n 0
Usingnotation:WN  e
k 
2
k
N
 X k ,k0,1,..., N  1
(4.21.1)
uniformly spaced frequency samples
,k0,1,..., N  1
(4.21.2)
Finite sum! Therefore, it’s computable.
j
2
N
(4.21.3)
N 1
(4.21.1) can be rewritten as:
X k   xnWN kn
(4.21.4)
n 0
1 N 1
IDFT :xn   X kWN  kn ,n  0,1,..., N  1
N n 0
Btw, DFT is a sampled version of DTFT.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.21.5)
22
DFT and IDFT (cont)
Let us verify (4.21.5). We multiply both sides by WN l n
N 1
x W
n 0
n
N
N 1
l n
1 N 1 N 1
 1 N 1
 kn 
l n
    X kWN WN   X kWN ( k l ) n
N n 0 k 0
n 0  N k 0

N 1
 xnWN l n
n 0
1 N 1 N 1  ( k l ) n
  X k WN
N k 0 n 0
 N , fork  l  rN ,risInteger
SinceWN ( k l ) n  
n 0
0,otherwise
(4.22.1)
(4.22.2)
N 1
(4.22.3)
N 1
 xnWN l n  X l
(4.22.4)
n 0
ELEN 5346/4304
DSP and Filter Design
Fall 2008
23
FFT
In the matrix form:
where:
X  DN xDFT
1
x  DN X IDFT
(4.23.2)
x[ x0 x1...xN 1 ]T timedomainsignal
(4.23.3)
X [ X 0  X1... X N 1 ]T NDFT samples
(4.23.4)
1
1
1
1 W 1
WN 2
N

DN  1 WN 2
WN 4


1 WN N 1 WN 2( N 1)
(4.23.5)

WN N 1 
WN 2( N 1) DFT matrix


( N 1)( N 1) 
WN

1
1
1
1 W 1
WN 2
N

1
DN 1  1 WN 2
WN 4
N

1 WN  ( N 1) WN 2( N 1)
1

WN  ( N 1) 
WN 2( N 1) IDFT matrix


 ( N 1)( N 1) 
WN

D 1
1
N
This is actually FFT…
ELEN 5346/4304
(4.23.1)
DSP and Filter Design
Fall 2008
(4.23.6)

1
DN *
N
(4.23.7)
24
Relation between DTFT and DFT
1. Sampling of DTFT
Letxn    X  e j 
DTFT
Sample X  e j atk  2
  
k
,k 0,1,...N  1N  frequencysamples X e jk  Yk N  point DFT
N
Yk    yn ,n0,1,..., N  1
IDFT

Yk  X e
jk

1 N 1
yn  YkWN  kn
N k 0
 j 2 Nk  
kl
 X e
   xlWN

 l 

1 N 1 
 1 N 1  k ( nl ) 
kl
 kn
   xlWN WN   xl  WN

N k 0 l 
l 
 N k 0

Considering yn 

x
m 
n  mN
,n0,1,..., N  1
(4.24.1)
(4.24.2)
(4.24.3)
yn is an infinite sum of shifted replicas of xn. Iff xn is a length M sequence (M  N)
than yn = xn. Otherwise, time-domain aliasing  xn cannot be recovered!
ELEN 5346/4304
DSP and Filter Design
Fall 2008
25
Relation between DTFT and DFT (cont)
2. DTFT from DFT by Interpolation

Let xn be a length N sequence: xn 
N  pointDFT
 X k
Let us try to recover DTFT from DFT (its sampled version).
N 1
X  e j    xne jn
n 0
N 1  j    2 k  n


N

Since e
n 0
N 1
N 1 j 2 kn
1 N 1  N 1
1

   X kWN  kn e jn   X k  e N e jn
N n 0  k 0
N k 0 n 0

1  e j ( N  2 k ) e j ( N  2 k )/2


1  e j (  2 k / N ) e j (  2 k )/2 N
(4.25.1)
  N  2 k 
  N  2 k 
sin 
sin


  j    2 k  N 1 
2
2



  e  N 2  (4.25.2)


  N  2 k 
  N  2 k 
sin 
 sin 

2N
2N




  N  2 k 
sin

  j    2 k  N 1  1
N 1
1
2

 e  N 2  
X  e j    X k
N k 0
N
  N  2 k 
sin 

2N


 N 
sin
N 1


N 1
2   j 2

Xk
e

1
 
k 0
sin  
2
(4.25.3)
It’s possible to determine DTFT X(ej) from its uniformly sampled version uniquely!
ELEN 5346/4304
DSP and Filter Design
Fall 2008
26
Relation between DTFT and DFT (cont 2)
3. Numerical computation of DTFT from DFT
Let xn is a length N sequence:
We wish to evaluate
X e
j

xn    X  e j 
DTFT
defined by N uniformly spaced samples
at more dense frequency scale.
k  2 k / M ,k  0,1,..., M 1,whereM


N 1
N 1
X e jk   xn e jk n   xne
n 0
Define: xl , n
 j 2 k
N
n
M
(4.26.1)
n 0
 xn ,0  n  N  1

0, N  n  M  1

 X e
jk
M 1
 x
n 0
l ,n
e
 j 2 k
zero-padding
n
M
M 1
  xl ,nWM kn
n 0
No change in information, no change in DTFT… just a better “plot resolution”.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.26.2)
(4.26.3)
27
A note on WN
Recallthat:WN  e
j
2
N
(4.27.1)
WN is also called an Nth root of unity, since
WN  e
j
Im
Im
2
N
Re
Re
n
ELEN 5346/4304
DSP and Filter Design
Fall 2008
2
N
2
N
1
(4.27.2)
28
DFT properties
1. Circular shift
xn is a length N sequence defined for n = 0,1,…N-1.
An arbitrary shift applied to xn will knock it out of the 0…N-1 range.
Therefore, a circular shift that always keeps the shifted sequence in
the range 0…N-1 is defined using a modulo operation:
xc ,n  x( n n0 moduloN  x nn0
N

 xnn0 ,n0  n  N  1


 xN n0  n ,0  n  n0
x n n0
N
N 1
xn
ELEN 5346/4304
x n1  x n5
6
DSP and Filter Design
6
x n4  x n 2
6
1

N
(4.28.1)
N 1
 k ( n  n0 )
X
W

 k N
k 0
(4.28.2)
1
kn0
kn0
 kn
W
X
W

W
Xk

N
k N
N
DFT
N k 0
6
Fall 2008
29
DFT properties (cont)
2. Circular convolution
A linear convolution for two length N sequences xn and gn has a length 2N-1:
N 1
yL,n   g m xn m ,0  n  2 N  2
(4.29.1)
m0
A circular convolution is a length-N sequence defined as:
N 1
yC ,n   gm x nm  gn xn xn gn
N
g
m 0
N 1
m
x nm
N
1 N 1
1 N 1 N 1
 k ( nm)
  g m  X kWN
  glWN km X kWN  kn
N k 0
N k 0 m0
m 0
x nm
N
1 N 1
  Gk X kWN  kn Gk X k
DFT
N k 0
Procedure: take two sequences of the same length (zero-pad if needed),
DFT of them, multiply, IDFT: a circular convolution.
ELEN 5346/4304
(4.29.2)
N
m 0
N 1
N
DSP and Filter Design
Fall 2008
(4.29.3)
30
DFT properties (cont 2)
Example:
X  e j    xne jn
(4.30.1)
n
Take N frequency samples of (4.30.1) and then IDFT:
2
2
N 1
 j kl
j


1 N 1
1
 kn
 kn
N
N
xn   X kWN   xl e
WN  WN  e

N k 0
N k 0 l


1 N 1  k ( n l )
  xl WN
  xn  rN
N k 0
l
r

(4.30.2)
aliased version of xn
 nl rN (4.22.3)
r
The results of circular convolution differ from the linear convolution “on the edges”
– caused by aliasing.
To avoid aliasing, we need to use zero-padding…
ELEN 5346/4304
DSP and Filter Design
Fall 2008
31
Linear filtering via DFT
Often, we need to process long data sequences; therefore, the input must
be segmented to fixed-size blocks prior LTI filtering. Successive blocks
are processed one at a time and the output blocks are fitted together…
We can do it by FFT: IFFT{FFT{x}FFT{h}}…
Problem: DFT implies circular convolution – aliasing!
yn  hn  xn
(4.31.1)
Assuming that hn is an M-sequence, we form an N-sequence (L - block
length):
xm,n
 xn mL  n  mL  L  1

0otherwise
N >> M; L >> M; N = L + M - 1 and is a power of 2
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(4.31.2)
32
Linear filtering via DFT (cont)
Next, we compute N-point DFTs of xm,n and hn, and form
Ym,k  Hk X m,k k  0,1,...N 1
Ym , k    ym ,n
IDFT
(4.32.1)
- no aliasing!
Since each data block was terminated
with M -1 zeros, the last M -1 samples
from each block must be overlapped
and added to first M – 1 samples of
the succeeded block.
An Overlap-Add method.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
33
Linear filtering via DFT (cont 2)
Alternatively:
Each input data block contains M -1
samples from the previous block followed
by L new data samples; multiply the NDFT of the filter’s impulse response and
the N-DFT of the input block, take IDFT.
Keep only the last L data samples from
each output block.
The first block is padded by M-1 zeros.
An Overlap-Save method.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
34
DFT properties: General
from Mitra’s
book
Btw, g[n] = gn
ELEN 5346/4304
DSP and Filter Design
Fall 2008
35
DFT properties: Symmetry
from Mitra’s
book
xn is a real
sequence
ELEN 5346/4304
DSP and Filter Design
Fall 2008
36
DFT properties: Symmetry (cont)
from Mitra’s
book
xn is a complex
sequence
ELEN 5346/4304
DSP and Filter Design
Fall 2008
37
N-point DFTs of 2 real sequences
via a single N-point DFT
Let gn and hn are two length N real sequences.
Form
xn = gn + jhn  Xk
(4.37.1)


1
X k  X * k
N
2
1
Hk 
X k  X * k
N
2j
Gk 

Notethat X *  k
ELEN 5346/4304
DSP and Filter Design
N
 X * N k
(4.37.2)

(4.37.3)
(4.37.4)
N
Fall 2008
38
Summary
Algorithm
Time
Frequency
CTFT
Continuous
Continuous
DTFT
Discrete
Continuous
DFT
Discrete
Discrete
ELEN 5346/4304
DSP and Filter Design
Fall 2008