Transcript DSP & Digital Filters

Spectral Analysis
• Spectral analysis is concerned with the
determination of the energy or power
spectrum of a continuous-time signal
g a (t )
• It is assumed that g a (t ) is sufficiently
bandlimited so that its spectral
characteristics are reasonably estimated
from those of its of its discrete-time
equivalent g[n]
1
Professor A G Constantinides
Spectral Analysis
• To ensure bandlimited nature g a (t ) is
initially filtered using an analogue antialiasing filter the output of which is
sampled to provide g[n]
• Assumptions:
(1) Effect of aliasing can be ignored
(2) A/D conversion noise can be neglected
2
Professor A G Constantinides
Spectral Analysis
• Three typical areas of spectral analysis are:
• 1) Spectral analysis of stationary sinusoidal
signals
• 2) Spectral analysis of of nonstationary
signals
• 3) Spectral analysis of random signals
3
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Assumption - Parameters characterising
sinusoidal signals, such as amplitude,
frequency, and phase, do not change with
time
• For such a signal g[n], the Fourier analysis
can be carried out by computing the DTFT

G ( e j )   g [ n ] e  j n
n  
4
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Initially the infinite-length sequence g[n] is
windowed by a length-N window w[n] to
yield  [n]
• DTFT (e j ) of  [n] then is assumed to
provide a reasonable estimate of G (e j )
• (e j ) is evaluated at a set of R ( R  N )
discrete angular frequencies using an Rpoint FFT
5
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
Note that
[k ]  (e j )
 2 k / R
, 0  k  R 1
• The normalised discrete-time angular
frequency corresponding to DFT bin k is
2

k
k 
R
• while the equivalent continuous-time
angular frequency is
2

k
 
6
k
RT Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Consider g[n]  cos(o n   ),    n  
• expressed as
g[n]  12 e j (o n )  e j (o n )


• Its DTFT is given by
j
G(e )  e
j
 e
7

  (  o  2)
  

 j
  (  o  2)
  
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• G (e j )is a periodic function of  with a period
2 containing two impulses in each period
• In the range       , there is an impulse at
  o of complex amplitude  e j and an
impulse at   o of complex amplitude  e j
• To analyse g[n] using DFT, we employ a finitelength version of the sequence given by
 [n]  cos(on   ), 0  n  N  1
8
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Example - Determine the 32-point DFT of a
length-32 sequence g[n] obtained by
sampling at a rate of 64 Hz a sinusoidal
signal g (t ) of frequency 10 Hz
a
• Since
Hz the DFT bins will be
F

64
T
located in Hz at ( k/NT)=2k, k=0,1,2,..,63
• One of these points is at given signal
frwquency of 10Hz
9
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• DFT magnitude plot
|[k]|
15
10
5
0
0
10
20
30
k
10
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Example - Determine the 32-point DFT of a
length-32 sequence [n] obtained by sampling at a
rate of 64 Hz a sinusoid of frequency 11 Hz
• Since
f R 11 32
FT

64
 5.5
the impulse at f = 11 Hz of the DTFT appear
between the DFT bin locations k = 5 and k = 6
• the impulse at f= -11 Hz appears between the DFT
bin locations k = 26 and k = 27
11
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• DFT magnitude plot
|[k]|
15
10
5
0
0
10
20
30
k
• Note: Spectrum contains frequency
components at all bins, with two strong
components at k = 5 and k = 6, and two
strong components at k = 26 and k = 27
12
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• The phenomenon of the spread of energy from a
single frequency to many DFT frequency locations
is called leakage
15
|[k]|
|(ej)|
10
5
0
0
10
20
30
k
13
• Problem gets more complicated if the signal
contains more than one sinusoid
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Example x[n]  1 sin(2 f n)  sin(2 f n),
1
2
2
0  n  N 1
0.34
• N - 16, f1  0.22, f2N= 16,
R = 16
6
|X[k]|
4
2
0
0
5
10
15
k
14
• From plot it is difficult to determine if there is one
or more sinusoids in x[n] and the exact locations of
the sinusoids
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
N = 16, R = 128
8
|X[k]|
6
4
2
0
0
50
100
k
• An increase in resolution
and accuracy of
the peak locations is obtained by increasing
DFT length to R = 128 with peaks occurring
at k = 27 and k =45
15
Professor A G Constantinides
Spectral Analysis of
Sinusoidal Signals
• Reduced resolution occurs when the
difference between the two frequencies
becomes less than 0.4
• As the difference between the two
frequencies gets smaller, the main lobes of
the individual DTFTs get closer and
eventually overlap
16
Professor A G Constantinides
Spectral Analysis of
Nonstationary Signals
• An example of a time-varying signal is the
2
x
[
n
]

A
cos(

n
chirp signal
o ) and shown
5
below for  o  10  10
Amplitude
1
0.5
0
-0.5
-1
0
100
200
300
400
500
Time index n
600
700
800
• The instantaneous frequency of x[n] is 2 o n
17
Professor A G Constantinides
Spectral Analysis of
Nonstationary Signals
18
• Other examples of such nonstationary
signals are speech, radar and sonar signals
• DFT of the complete signal will provide
misleading results
• A practical approach would be to segment
the signal into a set of subsequences of
short length with each subsequence centered
at uniform intervals of time and compute
DFTs of each subsequence
Professor A G Constantinides
Spectral Analysis of
Nonstationary Signals
• The frequency-domain description of the
long sequence is then given by a set of
short-length DFTs, i.e. a time-dependent
DFT
• To represent a nonstationary x[n] in terms of
a set of short-length subsequences, x[n] is
multiplied by a window w[n] that is
stationary with respect to time and move
x[n] through the window
19
Professor A G Constantinides
Spectral Analysis of
Nonstationary Signals
• Four segments of the chirp signal as seen
through a stationary length-200 rectangular
window
1
Amplitude
Amplitude
1
0
-1
0
50
100
150
Time index n
200
0
-1
100
20
0
-1
200
200
250
Time index n
300
350
400 A G450
500
Professor
Constantinides
Time index n
1
Amplitude
Amplitude
1
150
250
300
350
Time index n
400
0
-1
300
Short-Time Fourier Transform
• Short-time Fourier transform (STFT),
also known as time-dependent Fourier
transform of a signal x[n] is defined by
X STFT (e
j

, n)   x[n  m] w[m] e
 j m
m  
where w[n] is a suitably chosen window
sequence
• If w[n] = 1, definition of STFT reduces to
that
of
DTFT
of
x[n]
21
Professor A G Constantinides
Short-Time Fourier Transform
j
• X STFT (e , n) is a function of 2 variables:
integer time index n and continuous
frequency 
j
X
(
e
, n) is a periodic function of 
•
STFT
with a period 2
j
• Display of X STFT (e , n) is the
spectrogram
• Display of spectrogram requires normally
three dimensions
22
Professor A G Constantinides
Short-Time Fourier Transform
• Often, STFT magnitude is plotted in two
dimensions with the magnitude represented
by the intensity of the plot
• Plot of STFT magnitude of chirp sequence
x[n]  A cos( o n 2 ) with  o  10  105
for a length of 20,000 samples computed
using a Hamming window of length 200
shown next
23
Professor A G Constantinides
Short-Time Fourier Transform
0.5
Frequency
0.4
0.3
0.2
0.1
0
0
5000
10000
Time
15000
• STFT for a given value of n is essentially
the DFT of a segment of an almost
sinusoidal sequence
24
Professor A G Constantinides
Short-Time Fourier Transform
• Shape of the DFT of such a sequence is
similar to that shown below
• Large nonzero-valued DFT samples around
the frequency of the sinusoid
• Smaller nonzero-valued DFT samples at
other frequency points
20
Magnitude
15
25
10
5
0
0
0.5


1.5
2
Professor
A G Constantinides
STFT on Speech
• An example of a narrowband spectrogram
of a segment of speech signal
26
Professor A G Constantinides
STFT on Speech
• The wideband spectrogram of the speech signal
is shown below
• The frequency and time resolution tradeoff
between the two spectrograms can be seen
27
Professor A G Constantinides