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Chapter 5Ideal Filters, Sampling, and
ReconstructionSections 5.3-5.4
Wed. June 26, 2013
Chapter 5Ideal Filters, Sampling, and
ReconstructionSections 5.3-5.4
Wed. June 26, 2013
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• To avoid phase distortion:
5.3.1 Ideal Filters – Linear Phase5.3.1 Ideal Filters – Linear Phase
( ) dH t for all ω in the passband, where td is a fixed positive number. Then the response to a sinusoidal input is:
( ) ( ) cos( )
( ) cos[ ( )]
o o o d
o o d
y t A H t t
A H t t
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5.3.1 Ideal Filters – Linear Phase5.3.1 Ideal Filters – Linear Phase
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5.3.2 Linear Phase Low-Pass Filter5.3.2 Linear Phase Low-Pass Filter
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5.3.2 Linear Phase Low-Pass Filter5.3.2 Linear Phase Low-Pass Filter
( )H
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5.3.2 Linear Phase Low-Pass Filter5.3.2 Linear Phase Low-Pass Filter
Let’s do this in the time domain:
with 2B and using time-shift property:
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5.3.2 Linear Phase Low-Pass Filter5.3.2 Linear Phase Low-Pass Filter
Non-Causal !Impulse response starts before t = 0
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5.3.2 Linear Phase Low-Pass Filter5.3.2 Linear Phase Low-Pass Filter
• Ideal filters are non-causal, due to the unrealistic vertical cutoffs.
•Design of realistic causal filters will be addressed later.
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5.4 Sampling5.4 Sampling
Sampled signal:
( ) ( ) ( ) where ( ) train of delta functions.sx t x t p t p t
( ) ( ) ( )sn
x t x nT t nT
where T is the sampling interval, and n is the sample number.
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5.4 Sampling5.4 Sampling
Sampled signal:
( ) ( ) ( ) where ( ) train of delta functions.sx t x t p t p t
( ) ( ) ( )sn
x t x nT t nT
where T is the sampling interval, and n is the sample number.
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5.4 Sampling5.4 Sampling
As shown in text equations 5.49 – 5.51, the Fourier transform of the sampled signal is:
1( ) ( )
where ( ) is the transform of ( ),
2and is the sampling frequency
s sk
s
X X kT
X x t
T
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5.4 Sampling5.4 Sampling
( )X
( )SX
s B s B s
Original signal:
Sampled signal:
/A T
A
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5.4.1 Reconstruction5.4.1 Reconstruction
( )X
( )SX
s B s B s
Original signal:
Sampled signal:
Apply ideal lowpass filter with bandwidth between B and ωs-B.
5.4.1 Reconstruction5.4.1 Reconstruction
Sampling theorem: Original signal can be recovered perfectly as long as it is sampled above the Nyquist rate. The Nyquist rate is twice the bandwidth of the original signal:
2N B In practice, signals are oversampled, i.e.,
where 2.s rB r
5.4.2 Reconstruction5.4.2 Reconstruction
Interpolation (reconstruction) formula is derived from the impulse response of the ideal low-pass filter:
( ) ( )sinc ( )n
BT Bx t x nT t nT
This shows how to “connect the dots” between the samples to recover the original signal. Notice that every sample contributes to every time point. It is not as simple as connecting the dots with straight lines!
5.4.3 Aliasing5.4.3 Aliasing
If the sampling rate is below the Nyquist rate, aliasing occurs.
