Transcript Document
ECE 8443 – Pattern Recognition
LECTURE 18:
FOURIER ANALYSIS OF CT SYSTEMS
•
Objectives:
Response to a Sinusoidal Input Frequency Analysis of an RC Circuit Response to Periodic Inputs Response to Nonperiodic Inputs Analysis of Ideal Filters
•
Resources:
Wiki: The RC Circuit CN: Response of an RC Circuit CNX: Ideal Filters URL: Audio:
Response of an LTI System to a Sinusoid
• • • • •
Consider an LTI CT system with impulse response h(t):
y
(
t
)
h
(
t
) *
x
(
t
)
h
( )
x
(
t
)
d
We will assume that the Fourier transform of h(t) exists:
H
( )
h
(
t
)
e
j
t dt
The output can be computed using our Fourier transform properties:
Y
( )
H
( )
X
( ) and
Y
( )
H
( )
X
( )
Y
( )
H
( )
X
( )
Suppose the input is a sinusoid:
x
(
t
)
A
cos( 0
t
)
Using properties of the Fourier transform, we can compute the output:
X
( )
A
e
j
0
e j
0
Y
( )
y
(
t
)
F
H
( )
X A
H A
H
( ( ) (
e
) 0 )
j
e
A
A
-
1
H H
Y
( ( ( ) 0 0 ) )
e e
A
j j j
H
H
H
( 0 ( 0 ( 0 )
e
0 ) )
j
0
e
cos
j
0
t
H
0 ( 0 0 0 )
e e
j
e j
H j
H
(
H
( 0 ( 0 0 ) ) )
e
0
j
0 0
ECE 3163: Lecture 18, Slide 1
Example: RC Circuit
dy
(
t
)
dt
1
RC y
(
t
) 1
RC x
(
t
) •
Using our FT properties:
j
Y
1
RC Y
1
RC X
( )
Y
( )
H
( )
j
1 /
RC
1 /
RC Y X
( )
X j
1 /
RC
1 /
RC H
( ) 1 /
RC
H
( ) 2 tan 1 ( 1 /
RC
) 2
RC
•
Compute the frequency response: RC = 0.001; W=0:50:5000; H=(1/RC)./(j*w+1/RC); magH=abs(H); angH=180*angle(H)/pi; ECE 3163: Lecture 18, Slide 2
Example: RC Circuit (Cont.)
• • • We can compute the output for RC =0.001
y
(
t
)
A
( 0 .
707 ) cos 1000
t
45 We can compute the output for RC =0.001
y
(
t
)
A
( 0 .
316 ) cos 3000
t
71 .
6
and
0 =1000 rad/sec
: and
0 =3000 rad/sec
: Hence the circuit acts as a lowpass filter. Note the phase is not linear.
•
If the input was the sum of two sinewaves:
x
(
t
) cos 100
t
cos 3000
t
describe the output.
ECE 3163: Lecture 18, Slide 3
Response To Periodic Inputs
• • •
We can extend our example to all periodic signals using the Fourier series:
x
(
t
)
a
0
k
1
A k
cos
k
0
k
(a variant of the trigonome tric Fourier series)
The output of an LTI system is:
y
(
t
)
a
0
H
k
1
A k H
k
0 cos
k
0
t
k
H
k
0
We can write the Fourier series for the output as:
y
(
t
)
a
0
y
k
1
A k y
cos
k
0
t
k y
where, a y 0
a
0
x H
( 0 )
A k y
A k x H
(
k
0 )
k y
k x
H
(
k
0 ) also, c y k 1 2
A k x H
(
k
0 ) and c k y
k x
H
(
k
0 ) •
It is important to observe that since the spectrum of a periodic signal is a line spectrum, the output spectrum is simply a weighted version of the input, where the weights are found by sampling of the frequency response of the LTI system at multiples of the fundamental frequency,
0
.
ECE 3163: Lecture 18, Slide 4
Example: Rectangular Pulse Train and an RC Circuit
•
Recall the Fourier series for a periodic rectangular pulse:
x
(
t
)
a
0
k
1
a k
cos • where,
a
0 0 .
5
a k
sin (
k
k
/ / 2 ) 2
Also recall the system response was:
H
( )
j
1 /
RC
1 /
RC
•
The output can be easily written as:
y
(
t
)
a
0
y
k
1
A k y
cos 0
t
k y
where,
a
0 y
A k y
a
0
x H
( 0 ) 0 .
5
A k x H
(
k
0 ) sin (
k
k
/ / 2 ) 2 1 /
RC
(
k
) 2 ( 1 /
RC
) 2 2
k
1 /
RC
(
k
) 2 0 ( 1 /
RC
) 2
k
odd
k
even
ECE 3163: Lecture 18, Slide 5
Example: Rectangular Pulse Train (Cont.)
•
We can write a similar expression for the output:
y
(
t
)
a
0
y
k k
1
odd
2
k
1 /
RC
(
k
) 2 ( 1 /
RC
) 2 cos
k
t
tan 1
k
RC
1/
RC
= 1 •
We can observe the implications of lowpass filtering this signal.
•
What aspects of the input signal give rise to high frequency components?
•
What are the implications of increasing
1/ RC in the circuit? •
Why are the pulses increasingly rounded for lower values of
1/ RC? •
What causes the oscillations in the signal as
1/ RC is increased? 1/
RC
= 10 1/
RC
= 100
ECE 3163: Lecture 18, Slide 6
Response to Nonperiodic Inputs
• •
We can recover the output in the time domain using the inverse transform:
y
(
t
) 1 2
H
(
e j
)
X
(
e j
)
e j
t d
These integrals are often hard to compute, so we try to circumvent them using transform tables and combinations of transform properties.
•
Consider the response of our RC circuit to a single pulse:
X
(
e j
) sin( ( / / 2 ) 2 )
H
(
e j
)
Y
(
e j
)
j
1 /
RC
1 /
RC X
(
e j
)
H
(
e j
) sin( ( / / 2 ) 2 )
j
1 /
RC
1 /
RC
•
MATLAB code for the frequency response: RC=1; w=-40:.3:40; X=2*sin(w/2)./w; H=(1/RC)./(j*w+1/RC); Y=X.*H; magY=abs(Y); ECE 3163: Lecture 18, Slide 7
Response to Nonperiodic Inputs (Cont.)
•
We can recover the output using the inverse Fourier transform: syms X H Y y w X = 2*sin(w/2)./w; H=(1/RC)./(j*w+1/RC); Y=X.*H; Y=ifourier(Y); ezplot(y,[-1 5]); axis([-1 5 0 1.5])
1/
RC
= 1 1/
RC
= 1 1/
RC
= 10 1/
RC
= 10
ECE 3163: Lecture 18, Slide 8
Ideal Filters
•
The process of rejecting particular frequencies or a range of frequencies is called filtering. A system that has this characteristic is called a filter .
•
An ideal filter is a filter whose frequency response goes exactly to zero for some frequencies and whose magnitude response is exactly one for other ranges of frequencies.
• •
To avoid phase distortion in the filtering process, an ideal filter should have a linear phase characteristic. Why?
H
(
e j
)
t d
for all ω in the filter passband
We will see this “ideal” response has some important implications for the impulse response of the filter.
•
Lowpass
•
Highpass
•
Bandstop
•
Bandpass ECE 3163: Lecture 18, Slide 9
Ideal Linear Phase Lowpass Filter
•
Consider the ideal lowpass filter with frequency response:
H
(
e j
)
e
j
t d
0 ,
B
B
,
B B
•
Using the Fourier transform pair for a rectangular pulse, and applying the time-shift property:
h
(
t
)
B
sin
c B
(
t
t d
•
Is this filter causal?
•
The frequency response of an ideal bandpass filter can be similarly defined:
H
(
e j
)
e
j
t d
0 , ,
B
1
B
2 elsewhere •
Will this filter be physically realizable?
Why?
ECE 3163: Lecture 18, Slide 10
•
Phase Response
•
Impulse Response
Summary
•
Showed that the response of a linear LTI system to a sinusoid is a sinusoid at the same frequency with a different amplitude and phase.
•
Demonstrated how to compute the change in amplitude and phase using the system’s Fourier transform.
•
Demonstrated this for a simple RC circuit.
•
Generalized this to periodic and nonperiodic signals.
•
Worked examples involving a periodic pulse train and a single pulse.
•
Introduced the concept of an ideal filter and discussed several types of ideal filters.
•
Noted that the ideal filter is a noncausal system and is not physically realizable. However, there are many ways to approximate ideal filters, and that is a topic known as filter design.
ECE 3163: Lecture 18, Slide 11