Bài giảng Digital signal processing - Chapter 7: Frequency Analysis of Signals and Systems - Nguyễn Thanh Tuấn
Chapter 7
Frequency Analysis of Signals and Systems
Nguyen Thanh Tuan,M.Eng.
Department of Telecommunications (113B3)
Ho Chi Minh City University of Technology
Email: [email protected]om
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
Digital Signal Processing
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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 jn
n
where ω=2πf/fs
The spectrum X(w) is in general a complex-valued function of
frequency:
X() | X()| ej ()
where
() arg(X()) with - ()
: is the magnitude spectrum
: is the phase spectrum
| X()|
()
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Frequency analysis of signals and systems
Determine and sketch the spectra of the following signal:
a)
x(n) (n)
x(n) anu(n) with |a|<1
b)
is periodic with period 2π.
X()
X ( 2k) x(n)e j(2k)n x(n)e jn X()
n
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
, we can find the x(n) in time-
X()
domain as
1
x(n)
X ()ejnd
2
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.
Digital Signal Processing
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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,
(even symmetry)
(odd symmetry)
| X()|| X()|
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
E | x(n) |2
X () 2d
x
2
n
Sxx () | X()|2
is called the energy density spectrum of x(n)
Digital Signal Processing
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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) | x(n)e jn X ()
zej
n
F
x1(n) X1()
Linearity: if
F
x2 (n)X2 ()
F
a1x1(n)a2x2 (n)a1X1() a2 X2 ()
then
F
x(n) X()
Time-shifting: if
F
x(nk)e jk X()
then
Digital Signal Processing
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Frequency analysis of signals and systems
Properties of DTFT
F
x(n) X()
Time reversal: if
F
x(n)X()
then
F
x1(n) X1()
Convolution theory: if
F
x2 (n)X2 ()
F
x(n) x1(n)x2 (n)X() X1()X2 ()
then
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
Digital Signal Processing
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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
, 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 non-
rectangular 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
: c=1 for rectangular window and c=2 for Hamming
window.
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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
the formulation
Applying
, the minimum length L to
sinusoids show be 20
resolve all three
samples for the rectangular window, and L =40 samples for the
Hamming case.
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Frequency analysis of signals and systems
Example
Digital Signal Processing
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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)
is a continuous function of frequency and therefore, it is not a
X()
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:
L1
X () x(n)e jn 0 2
n0
2k
Sampling X(ω) at equally spaced frequency
where N ≥ L, we obtain N-point DFT of length L-signal:
, k=0, 1,…,N-1
k
N
L1
2k
X (k) X ( ) x(n)e j2kn/N
(N-point DFT)
N
n0
DFT presents the discrete-frequency samples of spectra of discrete-
time signals.
Digital Signal Processing
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Frequency analysis of signals and systems
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