Frequency Domain Representation of Discrete Time Signals & Systems
Let us assume we have an LTI system
x[n]
If
y[n]
h[.]
x[n]  e j 2f n
d
then


y[n]   h[k ]x[n  k ]   h[k ]e j 2f
k  
d
( nk )
k  

 e j 2f n  h[k ]e  j 2f k
d
d
k  
e
j 2 f d n
H (e jw )
Eigenfunction
Example:


Let x[ n ]  A cos 2fn   
A
2
eigenvalue

e
j   2fn
  e  j   2fn


y[ n ] 



 h[ k ] x[ n  k ]
k  

A


j   2f n  k
 h[ k ]e

2 k  
A
2
  
H e
jw
e
j   2fn
 
 
  H e  jw e  j   2fn
A special case of this problem exist when h[n] is real
 

 
H e jw  H e jw


 j   2f n  k
 h[ k ]e
k  

 


In this case
y[n]  A H e jw  cos2fn     
 
where   angH e jw
In other words a sinusoidal input to a discrete time LTI system provides a sinusoidal
output.
Frequency Domain Representation of Discrete-Time Signals and Systems
Discrete time Fourier Transform is a tool by which a time-domain sequence is mapped
into a continuous function of a frequency variable. Because the DTFT is periodic the
parent discrete-time sequence can be simply obtained by computing its Fourier Series
representation.
Definition of the Forward Transform
Discrete-time Fourier transform X e jw of a sequence x[n] is defined as:
 
   x[n]e
X e jw 

 jwn
(1)
n  
 
In general X e jw is a complex function of the real variable w and can be written as:
 
 
 
X e jw  X re e jw  jX im e jw
 
X e jw can alternatively be expressed in polar form as:
 
 
X e jw  X e jw e j ( w)
where,
  
( w)  arg X e jw
(2)
In many applications the Fourier transform is called the Fourier Spectrum and likewise
X e jw  and (w) are referred to as the “magnitude spectrum” and “phase spectrum”
respectively.
Note from eq.(2) that if we replace (w) with (w) + 2k , where k is an integer,
X e jw remains unchanged implying that the phase function cannot be uniquely specified
for any Fourier Transform.
 
We will assume from now on that unless otherwise specified (w) is restricted to a
2 range.
i.e 0≤ (w) < 2
-  ≤ (w) < 
Fourier transform of some sequences exhibit discontinuities of 2 in their phase
responses. The process of removing these discontinuities is called “unwrapping the
phase” and the new phase function will be denoted as  c (w) . The subscript “c” indicates
that phase is a continuous function of w.
Example:
Consider a causal sequence x[n] where;
x[n]  0.5 u[n]
n
 
Its DTFT X e jw can be obtained as
   0.5 u[n]e
X e jw 

n
n  


  0.5e jw
n 0

n

 jwn

  0.5 (1)e  jwn
n
n 0
1
1  0.5e  jw
Magnitude and phase of the above FT is shown below:
Note that the FT of the sequence x[n] is a CONTINUOUS function of ‘w’. It is also
periodic with period 2
Equation (1) therefore represents the Fourier Series representation of the periodic
function X e jw .
 
 
The Fourier coefficients x[n] can be computed from X e jw using the reverse Fourier
integral below;
1
x[n] 
2

 X e e
jw
jwn
dw

This is called the inverse discrete-time Fourier Transform (IDTFT).
Fourier Transform For Finite Sequences (DFT)
In practice the Fourier components of data are obtained by digital computation rather
than by analog processing. Hence the analog values have to be sampled at regular
intervals and the sample values are then converted to a digital binary representation. This
is done by using sample-and-hold circuit followed by ADC.
Provided that samples recorded per second is high enough the waveform will be
adequately represented. The necessary sampling rate is called the Nyquist rate and is
2 f max .
Let us assume that a waveform has been sampled at regular time intervals T to produce
the sample sequence:
xnT   x(0), x(T ),..............., xN 1T 
of N samples
where, n = 0,1,…….., (N-1).
The data values will be real only when representing the values of a time series such as a
voltage waveform.
The DFT of x(nT) is defined as the sequence of complex values :
X k  X (0), X (),..............., X N 1
where, Ω is the first harmonic frequency given by  
2
2

( N  1)T NT
for n  1
Note that X(kΩ) have real and imaginary parts for the kth harmonic
X(k) = R(k) + j I(k)
Here we have used X(k) to represent X(kΩ).
Note that N real data values (in time-domain) transforms to N complex DFT values (in
frequency domain).
The DFT values X(k) can be computed using:
N 1
X (k )  FD xnT )    x(nT )e  jknT
k  0,1,2,......., ( N  1)
n 0
here ,
FD denotes the discrete Fourier transformation
k represents the harmonic number of the transform component.

Let us compare this result to the continuous-time Fourier transform X ( jw)   x(t )e  jwt dt

When x(t) = 0 for T < 0 and t > (N-1)T the two equations can be analogous
x(nT) = x(t)
kΩ = w
nT = t
However, the two transform are NOT equal
If the above mentioned substitutions are made and we also put dt = T and replace the
integral by a summation, the harmonic frequencies kf s are:
fs 

1

2 N  1T
hence ;
N 1
 xnT e
 jknT
 T  F ( jw)
n 0

F ( jw)  T  X (k )
Example:
The DFT of the sequence { 1, 0, 0, 1} will be evaluated
x(0) = 1,
x(T)= 0,
x(2T) = 0, x(3T) = 1 ,
N=4
We desire to find X(k) for k = 0,1,2,3.
For k = 0
3
3
n 0
n 0
X (0)   x(nT )e j 0   x(nT )  x(0)  x(T )  x(2T )  x(3T )
 1 0  0 1  2
3
k=1
X (1)   x(nT )e
 jnT
n 0
3
  x(nT )e
 j 2n / N
 1  0  0  1e
j
6
4
 1 j
n 0
3
k=2
X (2)   x(nT )e  j 2n 2 / N  1  0  0  1e  j 3  1  1  0
n 0
3
k=3
X (3)   x(nT )e  j 2n 3 / N  1  0  0  1e
j
9
2
 1 j
n 0
Ans:
X(k) = { 2 , (1+j) , 0, (1-j) }
To find Ω it is necessary to know the value of T, the sampling interval
For example if the data sequence is sampled at 8kHz
T = 1/Fs = 1/8000= 125 s.
2
 12.57 1000rad / s
Then  
NT
,2  25.14 1000rad / s etc.
| X(k) |
2.0
2.0
2
2
0
12.57
25.14
37.71
50.28
k (1000)
rad/sec
(k)
45
0
12.57
25.14
37.71
50.28
k (1000)
rad/sec
- 45
We note that the amplitude plot is symmetrical about the second harmonic component,
that is about N/2.
Phase functions are an odd function centered around this point also.
An important property of the DFT may be deduced if the kth component of the DFT X(k),
is compared with the (k+N)th component X(k+N).
N 1
X (k )   x(nT )e
 jknT
n 0
  x(nT )e
 jk
2
n
N
N 1
N 1
n 0
n 0
X (k  N )   x(nT )e  jk 2 / N e  jN 2n / N   x(nT )e  jk 2 / N (1)
Hence we note that
X(k+N) = X(k)
This tells us that DFT is periodic with period N. This is known as the cyclical property of
DFT.
Definition of the IDFT
The IDFT is defined as:
x(nT )  F 1 X (k ) 
1 N 1
X (k )e jknT

N k 0
n  0,1,......., ( N  1)
Example:
For X(k) = { 1, 1+j , 0, 1-j} find the time-domain series X(nT)
1 N 1
 X (k )(1)
N k 0
1
 X (0)  X (1)  X (2)  X (3)
4
1
 2  1  j  0  1  j   1
4
x(0) 
For
n=0
n=1
n=2
x(T) = 0
x(2T) = 0
1
x(3) 
N
n=3
x(nT) = { 1 0 0
 X ( k )e
jk 3 / 2
k 0
1
 2  (1  j )e
4


N 1
1}
j 3
2

 (1  j )e j 9 / 2 

1
2  (1  j )(  j )  (1  j )( j )  1
4