Transcript Lecture 2: Continuation of Sampling

Lecture 2: Continuation of Sampling
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
 Butterworth Filters
The frequency response magnitude for an Nth order Butterworth
filter is of the form:
H0
H( f ) 
2N
 f 
1   
 f0 
–
–
The cutoff frequency is f0.
The roll-off rate is 20N dB/decade.
10-Sept., 1998
EE421, Lecture 2
1
Butterworth Filters
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
 2nd order Sallen-Key low-pass filter (can be cascaded to create
4th, 6th, 8th, … order filters)
C
+
-
+
R
R
+
(1-K)Rf
input
C
-
Rf
-
f0 
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output
1
2RC
EE421, Lecture 2
2
Butterworth Filters
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
 Example 2.1: Suppose H0 = 1 and you want to select f0 for 1 dB (or
less) attenuation over the band from 0 to 4 kHz.
attenuation ( dB )  20 log 10 H ( f )
  f 2 N 
 10 log 10 1    
  f 0  
 1
at f = 4 kHz. So,
or
4
 
 f0 
f0 
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2N
 100.1  1
4
10
0.1

1
1
2N
EE421, Lecture 2
in kHz
3
Ideal Reconstruction
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
signal spectrum
f
sampled-signal spectrum
ideal reconstruction filter
f
Reconstructed-signal spectrum
f
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EE421, Lecture 2
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Ideal Reconstruction
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
 The impulse response of an ideal reconstructor is a sinc function,
therefore an ideal reconstructor uses sinc interpolation.
sampled signal
Interpolation signals
Interpolated signal
10-Sept., 1998
EE421, Lecture 2
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Practical Reconstruction
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
 The staircase reconstructor is the simplest and most widely used
reconstructor in practice. Basically, it is a sample and hold.
sampled signal
Interpolation signals
The impulse response for a staircase
reconstructor is a rectangle. Therefore,
its frequency response is a sinc!
10-Sept., 1998
Interpolated signal
EE421, Lecture 2
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Staircase Reconstruction
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
signal spectrum
f
sampled-signal spectrum
staircase reconstruction filter
f
surviving replicas (images)
Reconstructed-signal spectrum
f
10-Sept., 1998
EE421, Lecture 2
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Staircase Reconstruction
EE421, Fall 1998
Michigan Technological University
Timothy J. Schulz
 The staircase reconstructor should be followed by a low-pass filter
to remove the residual images.
 Also, a staircase reconstructor suffers from nonuniform pass-band
attenuation. This can be corrected with a digital equalization
filter with a response equal to an inverse sinc function over the
pass-band.
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