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Chapter 7
Frequency Analysis of Signals and Systems
Nguyen Thanh Tuan, Click
M.Eng.
to edit Master subtitle style
Department of Telecommunications (113B3)
Ho Chi Minh City University of Technology
Email:


 Frequency analysis of signal involves the resolution of the signal into
its frequency (sinusoidal) components. The process of obtaining the
spectrum of a given signal using the basic mathematical tools is
known as frequency or spectral analysis.
 The term spectrum is used when referring the frequency content of a
signal.
 The process of determining the spectrum of a signal in practice base
on actual measurements of signal is called spectrum estimation.

 The instruments of software programs
used to obtain spectral estimate of such
signals are kwon as spectrum analyzers.

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Frequency analysis of signals and systems


 The frequency analysis of signals and systems have three major uses

in DSP:

1) The numerical computation of frequency spectrum of a signal.
2) The efficient implementation of convolution by the fast Fourier
transform (FFT)

3) The coding of waves, such as speech or pictures, for efficient
transmission and storage.

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Frequency analysis of signals and systems


Content

1. Discrete time Fourier transform DTFT
2. Discrete Fourier transform DFT

3. Fast Fourier transform FFT

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Transfer functions
and Digital Filter Realizations

1. Discrete-time Fourier transform (DTFT)
 The Fourier transform of the finite-energy discrete-time signal x(n) is
defined as:

X ( )   x(n)e jn
n 

where ω=2πf/fs
 The spectrum X(w) is in general a complex-valued function of
frequency:
X ( ) | X () | e j ( )

where  ()  arg( X ()) with -   ()  
 | X ( ) |
  ( )

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: is the magnitude spectrum
: is the phase spectrum

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Frequency analysis of signals and systems


 Determine and sketch the spectra of the following signal:
a) x(n)   (n)
b) x(n)  a nu(n) with |a|<1

 X ( ) is periodic with period 2π.
X (  2 k ) 





x ( n) e

 j (  2 k ) n

n 







x(n)e jn  X ( )

n 

The frequency range for discrete-time signal is unique over the
frequency interval (-π, π), or equivalently, (0, 2π).
 Remarks: Spectrum of discrete-time signals is continuous and
periodic.
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Frequency analysis of signals and systems


Inverse discrete-time Fourier transform (IDTFT)
 Given the frequency spectrum X ( ) , we can find the x(n) in timedomain as
1
x ( n) 
2





X ( )e jn d



which is known as inverse-discrete-time Fourier transform (IDTFT)
Example: Consider the ideal lowpass filter with cutoff frequency wc.
Find the impulse response h(n) of the filter.

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Frequency analysis of signals and systems


Properties of DTFT

 Symmetry: if the signal x(n) is real, it easily follows that
X  ( )  X ( )

or equivalently, | X () || X () |

(even symmetry)
(odd symmetry)
arg( X ())   arg( X ())

We conclude that the frequency range of real discrete-time signals can
be limited further to the range 0 ≤ ω≤π, or 0 ≤ f≤fs/2.
 Energy density of spectrum: the energy relation between x(n) and
X(ω) is given by Parseval’s relation:


1
2
E x   | x ( n) | 
2
n 





X ( ) d
2



S xx ( ) | X ( ) |2 is called the energy density spectrum of x(n)
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Frequency analysis of signals and systems


Properties of DTFT
 The relationship of DTFT and z-transform: if X(z) converges for

|z|=1, then
X ( z ) |z e    x(n)e jn  X ( )
j

n 

 Linearity: if

F
x1 (n) 
 X1 ( )

F
x2 (n) 
 X 2 ( )
F
a1 x1 (n)  a2 x2 (n) 
 a1 X1 ()  a2 X 2 ()

then

 Time-shifting: if
then

F
x(n) 
 X ( )

F
x(n  k ) 
 e jk X ( )

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Frequency analysis of signals and systems


Properties of DTFT
F
 X ( )
 Time reversal: if x(n) 
F
x(n) 
 X ( )

then

F
 X1 ( )
 Convolution theory: if x1 (n) 
F
x2 (n) 
 X 2 ( )

then

F
x(n)  x1 (n)  x2 (n) 
 X ()  X1 () X 2 ()

Example: Using DTFT to calculate the convolution of the sequences
x(n)=[1 2 3] and h(n)=[1 0 1].

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Frequency analysis of signals and systems


Frequency resolution and windowing
 The duration of the data record is:

 The rectangular window of length
L is defined as:

 The windowing processing has two major effects: reduction in the

frequency resolution and frequency leakage.
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Frequency analysis of signals and systems


Rectangular window

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Frequency analysis of signals and systems


Impact of rectangular window
 Consider a single analog complex sinusoid of frequency f1 and its
sample version:

 With assumption

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, we have

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Frequency analysis of signals and systems

Double sinusoids

 Frequency resolution:
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Frequency analysis of signals and systems


Hamming window

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Frequency analysis of signals and systems


Non-rectangular window
 The standard technique for suppressing the sidelobes is to use a nonrectangular window, for example Hamming window.
 The main tradeoff for using non-rectangular window is that its
mainlobe becomes wider and shorter, thus, reducing the frequency
resolution of the windowed spectrum.
 The minimum resolvable frequency difference will be

where
window.

Digital Signal Processing

: c=1 for rectangular window and c=2 for Hamming

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Frequency analysis of signals and systems


Example
 The following analog signal consisting of three equal-strength
sinusoids at frequencies

where t (ms), is sampled at a rate of 10 kHz. We consider four data
records of L=10, 20, 40, and 100 samples. They corresponding of the
time duarations of 1, 2, 4, and 10 msec.
 The minimum frequency separation is
Applying
the formulation
, the minimum length L to
resolve all three
sinusoids show be 20
samples for the rectangular window, and L =40 samples for the
Hamming case.
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Frequency analysis of signals and systems

Example

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Frequency analysis of signals and systems


Example

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Frequency analysis of signals and systems


2. Discrete Fourier transform (DFT)
 X ( ) is a continuous function of frequency and therefore, it is not a
computationally convenient representation of the sequence x(n).
 DFT will present x(n) in a frequency-domain by samples of its
spectrum X ( ) .
 A finite-duration sequence x(n) of length L has a Fourier transform:
L 1

X ( )   x(n)e jn

0    2

n 0

Sampling X(ω) at equally spaced frequency k  2 k , k=0, 1,…,N-1
where N ≥ L, we obtain N-point DFT of length N
L-signal:
L 1
2 k
X (k )  X (
)   x(n)e j 2 kn / N
(N-point DFT)
N
n 0

 DFT presents the discrete-frequency samples of spectra of discretetime signals.
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Frequency analysis of signals and systems


2. Discrete Fourier transform (DFT)
 With the assumption x(n)=0 for n ≥ L, we can write
N 1

X (k )   x(n)e j 2 kn / N , k  0,1,

, N  1.

(DFT)

n 0

 The sequence x(n) can recover form the frequency samples by inverse
DFT (IDFT)
1 N 1
x(n)   X (k )e j 2 kn / N , n  0,1,
N n 0

, N  1.

(IDFT)

Example: Calculate 4-DFT and plot the spectrum of x(n)=[1 1, 2, 1]

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Frequency analysis of signals and systems


Matrix form of DFT
 By defining an Nth root of unity WN  e j 2 / N , we can rewritte DFT
and IDFT as follows
N 1

X (k )   x(n)WNkn , k  0,1,

, N  1.

(DFT)

n 0

1 N 1
x(n)   X (k )WN kn , n  0,1,
N n 0

, N  1.

(IDFT)

 Let us define:  x(0) 

 X (0) 
 x(1) 
 X (1) 
 X 

xN  
N









x
(
N

1)
X
(
N

1)





The N-point DFT can be expressed in matrix form as: XN  WN x N
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Frequency analysis of signals and systems


Matrix form of DFT
1
1
1
1 W

2
W
N
N

WN  1 WN2
WN4


1 WNN 1 WN2( N 1)


WNN 1 
WN2( N 1) 


WN( N 1)( N 1) 
1

 Let us define:  x(0) 

 X (0) 
 x(1) 
 X (1) 
 X 

xN  
N









x
(
N

1)
X
(
N

1)





The N-point DFT can be expressed in matrix form as: XN  WN x N
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Frequency analysis of signals and systems


 Example: Determine the DFT of the four-point sequence x(n)=[1 1,

2 1] by using matrix form.

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Frequency analysis of signals and systems


Properties of DFT
Properties

Time domain

Frequency domain

 Notation

x ( n)

X (k )

 Periodicity
 Linearity

x(n  N )  x(n)

X (k )  X (k  N )

a1 x1 (n)  a2 x2 (n)

a1 X1 (k )  a2 X 2 (k )

 Circular time-shift

e j 2 kl / N X (k )

x((n  l )) N

 Circular convolution
 Multiplication
of two sequences

 Parveval’s theorem

Digital Signal Processing

1 N 1
Ex   | x(n) |   | X (k ) |2
N k 0
n 0
N

2

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Frequency analysis of signals and systems