lecture 4

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Husheng Li, UTK-EECS, Fall 2012
DISCRETE-TIME SIGNAL PROCESSING
LECTURE 4 (SAMPLING)
PERIODIC SAMPLING
๏ƒ’
Sampling: ๐‘ฅ ๐‘› = ๐‘ฅ๐‘ (๐‘›๐‘‡), where T is the
sampling period. In practice, it is done by A/D
converter. The sampling operation is generally
invertible.
TWO STAGE REPRESENTATION
We represent the sampling
procedure in two stages:
• Multiplication with an impulse
train ๐‘  = ∞
−∞ ๐›ฟ(๐‘ก − ๐‘›๐‘‡) with
output ๐‘ฅ๐‘  ๐‘ก = ๐‘ฅ๐‘ ๐‘ก ๐‘  ๐‘ก .
• Conversion from impulse train
to discrete time sequence
Note: this is a mathematical
formulation, not a physical circuit
implementation
FREQUENCY-DOMAIN REPRESENTATION
๏ƒ’
The frequency domain of the
post-sampling signal is given
by
๐‘‹๐‘  ๐‘—๐‘ค
∞
1
=
๐‘‹๐‘ (๐‘—(๐‘ค − ๐‘˜๐‘ค๐‘  ))
๐‘‡
๐‘˜=−∞
๏ƒ’
๏ƒ’
Assume that the signal has a
limited band −๐‘ค๐‘ , ๐‘ค๐‘ .
If the sampling frequency
satisfies ๐‘ค๐‘  ≥ 2๐‘ค๐‘ , there will
be no overlap.
EXACT RECOVERY
๏ƒ’
An ideal low pass
filter can be used to
obtain the exact
original signal.
ALIASING
If the inequality ๐‘ค๐‘ 
≥ 2๐‘ค๐‘ is not valid,
the frequency copies
of signal will overlap,
which incurs a
distortion called
aliasing.
๏ƒ’ See the example of
cosine function.
๏ƒ’
NYQUIST-SHANNON THEOREM
๏ƒ’
Theorem: For a band limited signal within band
−๐‘ค๐‘ , ๐‘ค๐‘ , it is uniquely determined by its
samples ๐‘ฅ๐‘ (๐‘›๐‘‡), if ๐‘ค๐‘  ≥ 2๐‘ค๐‘ .
EXAMPLE OF SINUSOIDAL SIGNAL
RECONSTRUCTION OF A BANDLIMITED SIGNAL
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The reconstruction is given
by
๐‘ฅ๐‘Ÿ ๐‘ก
∞
=
๐‘›=−∞
sin(๐œ‹ ๐‘ก − ๐‘›๐‘‡ /๐‘‡)
๐‘ฅ(๐‘›)
๐œ‹ ๐‘ก − ๐‘›๐‘‡ /๐‘‡
INTUITIVE EXPLANATION
It can be used for D/C
converter:
๐‘‹๐‘Ÿ ๐‘—๐‘ค = ๐ป๐‘Ÿ ๐‘—๐‘ค ๐‘‹(๐‘—๐‘ค)
๏ƒ’
DISCRETE-TIME PROCESSING
๏ƒ’
We can use C/D converter to convert a
continuous-time signal to a discrete-time one,
process it in a discrete-time system, and then
convert it back to continuous time domain.
EXAMPLE: LTI AND LPF
๏ƒ’
We can use a discrete-time low pass filter (LPF)
to do the low pass filtering for continuous time
signal.
EXAMPLE: LTI AND LPF
๏ƒ’
The ideal low pass
discrete-time filter
with discrete-time
cutoff frequency w
has the effect of an
ideal low pass filter
with cutoff
frequency w/T.
CONTINUOUS-TIME PROCESSING OF DISCRETETIME SIGNALS
๏ƒ’
We can also use continuous-time system to
process discrete-time signals.
RESAMPLING: DOWNSAMPLING
๏ƒ’
The downsampling ๐‘ฅ๐‘‘ ๐‘› = ๐‘ฅ(๐‘›๐‘€) implies
1
๐‘‹๐‘‘ ๐‘—๐‘ค =
๐‘€
๐‘€−1
๐‘–=0
๐‘ค
๐‘‹(๐‘—( − 2๐œ‹๐‘–/๐‘€))
๐‘€
INTUITION IN THE FREQUENCY DOMAIN
With
aliasing
Without
aliasing
DECIMATOR
๏ƒ’
A general system for downsampling by a factor
of M is the one shown above, which is called a
decimator.
UPSAMPLING
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The upsampling is given by ๐‘ฅ๐‘– ๐‘› = ๐‘ฅ(๐‘›/๐ฟ),
where L is the integer factor.
EXPANDER
The output of
expander is given
by ๐‘ฅ๐‘’ ๐‘›
= ∞
−∞ ๐‘ฅ(๐‘˜)๐›ฟ(๐‘› − ๐‘˜๐ฟ
.
๏ƒ’ In the frequency,
we have ๐‘‹๐‘’ ๐‘ค
= ๐‘‹ ๐‘ค๐ฟ .
๏ƒ’
INTERPOLATOR
๏ƒ’
It can be shown that the above structure realizes the
upsampling and interpolates the signals between samples:
∞
sin(๐œ‹(๐‘› − ๐‘˜๐ฟ)/๐ฟ)
๐‘ฅ๐‘– ๐‘› =
๐‘ฅ(๐‘˜)
๐œ‹(๐‘› − ๐‘˜๐ฟ)/๐ฟ
−∞
SIMPLE AND PRACTICAL INTERPOLATION
๏ƒ’
The ideal interpolator is impossible to implement.
In practice, we can use a linear interpolator:
๐‘›
โ„Ž ๐‘› = 1 − ๐ฟ , |๐‘›| ≤ ๐ฟ
0, ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’
TIME AND FREQUENCY OF LINEAR
INTERPOLATOR
CHANGING SAMPLING RATE BY A NON-INTEGER
FACTOR
๏ƒ’
The change of sampling rate by a non-integer factor can be
realized by the cascade of interpolator and decimator.
THE FREQUENCY INTUITION
MULTIRATE SIGNAL PROCESSING
๏ƒ’
Multirate techniques refer in general to utilizing
upsampling, downsampling, compressors and
expanders in a variety of ways to improve the
efficiency of signal processing systems.
INTERCHANGE OF FILTERING WITH
COMPRESSOR / EXPANDER
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The operations of linear filtering and
downsampling / upsampling can be exchanged
if we modify the linear filter.
MULTISTAGE DECIMATION
๏ƒ’
๏ƒ’
The two stage implementation is often much more
efficient than a single-stage implementation.
The same multistage principles can also be applied to
interpolation
DIGITAL PROCESSING OF ANALOG SIGNALS
๏ƒ’
In practice, continuous time signals are not
precisely band limited, ideal filters cannot be
realized, ideal C/D and D/C converters can only be
approximated by A/D and D/A converters.
PREFILTERING TO AVOID ALIASING
๏ƒ’
We can use oversampled A/D to simplify the
continuous-time antialiasing filter.
FREQUENCY DOMAIN INTUITION
๏ƒ’
Key point: the
noise is aliased;
but the signal is
not. Then, the
noise can be
removed using a
sharp-cutoff
decimation filter.
A/D CONVERSION
SAMPLE-AND-HOLD
The zero-order-hold
system has the
impulse response
given by
1, 0 < ๐‘ก < ๐‘‡
โ„Ž0 ๐‘ก =
0, ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’
๏ƒ’
QUANTIZATION
This quantizer
is suitable for
bipolar signals.
๏ƒ’ Generally, the
number of
quantization
levels should be
a power of tow,
but the number
is usually much
larger than 8.
๏ƒ’
ILLUSTATION
D/A CONVERSION
๏ƒ’
The ideal D/A is given by
∞
sin(๐œ‹ ๐‘ก − ๐‘›๐‘‡ /๐‘‡)
๐‘ฅ๐‘Ÿ ๐‘ก =
๐‘ฅ(๐‘›)
๐œ‹ ๐‘ก − ๐‘›๐‘‡ /๐‘‡
๐‘›=−∞
In practice, we need to use the above structure.
OVERSAMPLING
๏ƒ’
๏ƒ’
Oversampling can make it possible to implement sharp cutoff
antialiasing filtering by incorporating digital filtering and decimation.
Oversampling and subsequent discrete-time filtering and
downsampling also permit an increase in the step size of the
quantizer, or equivalently, a reduction in the number of bits required
in the A/D conversion.
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