EEO 401
Digital Signal Processing
Prof. Mark Fowler
Note Set #16
• Bandpass Sampling
• Reading Assignment: Sect. 6.4.1 of Proakis & Manolakis
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BP-Sampling of RF Signals: Basic Idea
Let fH = 3B
(This Example Only)
X( f )
Need Fs = 2B
B = fH – fL
fL fH
f
X( f±Fs )
f
X( f±2Fs )
f
X( f±3Fs )
f
Xs( f )
…
…
–Fs/2
Fs/2
f
BP Sampled Signal is a
Down-Shifted Version
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of the RF signal!!
BP-Sampling: Simple Case (fH = kB, k integer)
Consider the case where fH = kB (k an Odd Integer)
X( f )
B B B B
-fH -fL
B
fL fH
k=5 for this case
f
Whenever fH = kB, we can choose Fs = 2B to perfectly
“interweave” the shifted spectral replicas
Xs( f )
…
…
-Fs/2 Fs/2
fL fH
k is Odd Here
f
3/12
BP-Sampling: Simple Case (Cont.)
Consider the case where fH = kB (k an Even Integer)
X( f )
B
B
B
B
k=6 for this case
B B
fL f H
f
Whenever fH = kB, we can choose Fs = 2B to perfectly
“interweave” the shifted spectral replicas
Xs( f )
…
…
-Fs/2 Fs/2
k is Even Here
f
Note: If k is EVEN the spectrum in the 0 to Fs/2 range is flipped.
This is not usually a problem since the next step after BP
sampling is usually to create the lowpass equivalent signal, which
can be done in a way that gives either spectral orientation.
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BP-Sampling: General Case ( fH≠kB)
≤
2 f H ≤ kFs
( k − 1) Fs ≤ 2 f L
2 fH
2 fL
≤ Fs ≤
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k
k −1
To find the required value of k… re-write as:
k
1
≤
Fs 2 f H
2 f H ≤ kFs
( k − 1) Fs ≤ 2 f L
( k − 1) Fs ≤ 2( f H − B )
(A)
(B)
Now… solve these for k:
Multiply (A) times (B) gives (left x left & right x right):
1
 k 
[(k − 1) Fs ]   ≤ [ 2( f H − B )]  
 2 fH 
 Fs 
kB
≤1
fH
fH
k≤
B
kB
k −1 ≤ k −
fH
k max
 fH 
≤ 
B
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Resulting conditions needed for aliasing-free BP Sampling:
2 fH
2( f H − B )
fH
≤ Fs ≤
and k ≤
k
k −1
B
7/12
Using Plot to Visualize Allowed Fs…
Given fH and B… draw vertical line at fH/B
Dangerous place
to operate!!!
8/12
Using Guard Band Idea to Ease Sensitivity to “Change”
f L=′ f L − ∆BL
f H′= f L + ∆BH
B′= B + ( ∆BL + ∆BH )
Operating Point
must allow this
range to fit
Resulting
Tolerance Range
for Fs
9/12
Advantage of BP Sampling
A BP-Sampling ADC works like a
Mixer and a Baseband-sampling ADC
x(t)
Bandpass
Sampling
ADC
x(t)
LPF
BP Sampling
Reduces the
“Parts Count”
Baseband
Sampling
ADC
cos(2πfct)
10/12
ADC Specs: Sample Rate & ADC BW
• Sampling Rate
– Fastest Rate at which the ADC can be run
– Determines the Widest Signal Bandwidth that ADC can handle
• ADC Bandwidth
– Highest Frequency that ADC’s internal electronics can pass
– Determines Frequency Band ADC can handle (e.g. HF, UHF, VHF, etc.)
– Crucial for Undersampling Applications
Baseband Sampling
Low ADC BW
Low ADC Fs
f
Low ADC BW
High ADC Fs
f
Bandpass Sampling
High ADC BW
Low ADC Fs
f
High ADC BW
High ADC Fs
f
Narrow
Band
Wide
Band
11/12
www.harris.com
ADC’s Fs
ADC’s
BW
12/12