DIGITAL SIGNAL PROCESSING
DISCRETE TIME FOURIER
TRANSFORM
Road Map
Fourier Transform
Continuous –Time Fourier Transform
Discrete-Time Fourier Transforms
DTFT Properties
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Fourier Transform
• The Fourier series named after Joseph
Fourier decomposes a periodic function
into a sum of simple oscillating functions
namely sines and cosines
• A Fourier transform, transforms a time
domain sequence into a frequency domain
sequence.
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Fourier Transform
• The Fourier transform defines a
relationship between a signal in the time
domain and its representation in the
frequency domain. Being a transform, no
information is created or lost in the
process, so the original signal can be
recovered from knowing the Fourier
transform, and vice versa.
• The Fourier transform of a signal is a
continuous complex valued signal capable
of representing real valued or complex
valued continuous time signals.
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Fourier Transform
• The Fourier transform is defined by the
equation

X(f ) 
 j 2ft


 x t e dt

where X(f) is the Fourier transform of x(t) .
• Frequency is measured in Hertz, with f as the
frequency variable.
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Continuous-Time Fourier Transform
• The CTFT of a continuous time signal
xa (t ) is given by

X a ( j ) 
 jt
x
(
t
)
e
dt
a


• Often referred to as the Fourier
Spectrum or simply the spectrum of the
continuous time signal
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Inverse CTFT
• The inverse CTFT of a Fourier
transform X a ( j) is given by
1
xa (t ) 
2

jt
X
(
j

)
e
d

a


• Often referred to as the Fourier
Integral
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Continuous-Time Fourier Transform
•
 is real and denotes the continuous-
time angular frequency variable in radians
• In general the CTFT is a complex
function of  in the range      
• It can be expressed in the polar form as
X a ( j)  X a ( j) e
where
j a ( )
 ()  arg{ X a ( j)}
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Continuous-Time Fourier Transform
• The quantity X a ( j) is called the
magnitude spectrum and the quantity
 a () is called the phase spectrum
• Both the spectrums are real functions of

• In general the CTFT X a ( j) exists if
xa (t ) satisfies the Dirichlet conditions
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Continuous-Time Fourier Transform
Dirichlet Conditions
• The signal xa (t ) has a finite number of
discontinuities and a finite number of
maxima and minima in any finite interval
• The signal is absolutely integrable i.e.


xa t dt  

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Energy Density Spectrum
• The total energy E x of a finite energy
continuous-time complex signal xa (t ) is given by

Ex 
x
2
a
(t ) dt




x
(
t
)
x
 a a (t )dt

• The above expression can be rewritten as
 1
E x   xa (t ) 

 2


X


a
( j)e
 jt

d dt

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Energy Density Spectrum
• Interchanging the order of the integration we get



1

 jt
Ex 
X a ( j)   xa (t )e dt  d

2 



1

2
1

2

X

a
( j)X a ( j)d


X
( j) d
2
a

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Energy Density Spectrum
• Hence

1
 x(t ) dt  2
2

X
( j) d
2
a

• The above relation is more commonly
known as the Parseval's relation for
finite energy continuous-time signals
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Energy Density Spectrum
• The quantity X a ( j) 2
is called the energy
density spectrum of xa (t ) and usually denoted as
S xx ()  X a ( j)
2
• The energy over a specified range of frequencies
 a    b can be computed using
E x ,r
1

2
b
S
xx
()d
a
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Band-limited Continuous–Time
Signals
• A full-band, finite-energy, continuous –
time signal has a spectrum occupying
the whole frequency range      
• A band-limited continuous-time signal
has a spectrum that is limited to a
portion of the frequency range      
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Band-limited Continuous–Time
Signals
• An ideal band-limited signal has a
spectrum that is zero outside a finite
frequency range a    b that
is
0, 0     a

X a ( j)  

0, b    
• However an ideal band-limited signal
cannot be generated in practice
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Band-limited Continuous–Time
Signals
• Band-limited signals are classified
according to the frequency range where
most of the signal’s is concentrated
• A lowpass continuous-time signal has a
spectrum occupying the frequency
range    p   where  p is
called the bandwidth of the signal
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Band-limited Continuous–Time
Signals
• A highpass continuous-time signal has
a spectrum occupying the frequency
range 0   p     where
the bandwidth of the signal is from  p
to 
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Band-limited Continuous–Time
Signals
• A bandpass continuous-time signal has
a spectrum occupying the frequency
range 0  L    H  
where H  L is the bandwidth
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Discrete Time Fourier Transform
• Definition : The discrete-time Fourier
transform (DTFT) X (e j ) of a
sequence x[n] is given by
j
X (e ) 

 x[n]e
 jn
n  
j
• In general X (e ) is a complex function
of the real variable  and can be
written as X (e j )  X (e j )  X (e j )
re
im
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Discrete Time Fourier Transform
j
• X re (e ) and X im (e j ) are
respectively the real and imaginary parts
of X (e j ) and are real functions of 
•
j
X (e ) can alternatively be expressed
as
X (e )  X (e ) e
where
 ( )  arg{ X (e j )}
j
j
j (  )
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Discrete Time Fourier Transform
•
•
•
•
j
X (e ) is called the magnitude function
 ( ) is called the phase function
Both quantities are again real functions of 
In many applications the DTFT is called the
Fourier Spectrum
• Likewise X (e j ) and  ( ) are called
the magnitude and phase spectra
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Discrete Time Fourier Transform
• For a real sequence x[n] , X (e j ) and
X re (e j ) are even functions of  whereas  ( )
and X im (e j ) are odd functions of 
Note: X (e j )  X (e j ) e j (  2k )
j
 X (e ) e
j (  )
for any integer k
The phase function  ( ) cannot be uniquely
specified for any DTFT
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Discrete Time Fourier Transform
• Unless otherwise stated we shall assume
that the phase function  ( )
is
restricted to the following range of values :
    ( )  
called the principal value
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Discrete Time Fourier
Transform
• The DTFT of some sequences exhibit
discontinuities of 2 in the phase
response
• An alternate type of phase function that is
continuous function of  is often
used.
• It is derived from the original phase
function by removing the discontinuities
of 2
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Discrete Time Fourier
Transform
• The process of removing the
discontinuities is called wrapping
• The continuous phase function
generated by unwrapping is denoted
as c ( )
• In some cases discontinuities of 
may be present after unwrapping
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DTFT Properties
• There are number of important properties
of the DTFT that are useful in signal
processing applications
• These are listed here without proof
• Their proofs are quite straightforward
• We illustrate the applications of some of
the DTFT properties
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