Husheng Li, UTK-EECS, Fall 2012
DISCRETE-TIME SIGNAL PROCESSING
LECTURE 4 (SAMPLING)
PERIODIC SAMPLING

Sampling: 𝑥 𝑛 = 𝑥𝑐 (𝑛𝑇), 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 𝑠 = ∞
−∞ 𝛿(𝑡 − 𝑛𝑇) with
output 𝑥𝑠 𝑡 = 𝑥𝑐 𝑡 𝑠 𝑡 .
• Conversion from impulse train
to discrete time sequence
Note: this is a mathematical
formulation, not a physical circuit
implementation
FREQUENCY-DOMAIN REPRESENTATION

The frequency domain of the
post-sampling signal is given
by
𝑋𝑠 𝑗𝑤
∞
1
=
𝑋𝑐 (𝑗(𝑤 − 𝑘𝑤𝑠 ))
𝑇
𝑘=−∞


Assume that the signal has a
limited band −𝑤𝑁 , 𝑤𝑁 .
If the sampling frequency
satisfies 𝑤𝑠 ≥ 2𝑤𝑁 , there will
be no overlap.
EXACT RECOVERY

An ideal low pass
filter can be used to
obtain the exact
original signal.
ALIASING
If the inequality 𝑤𝑠
≥ 2𝑤𝑁 is not valid,
the frequency copies
of signal will overlap,
which incurs a
distortion called
aliasing.
 See the example of
cosine function.

NYQUIST-SHANNON THEOREM

Theorem: For a band limited signal within band
−𝑤𝑁 , 𝑤𝑁 , it is uniquely determined by its
samples 𝑥𝑐 (𝑛𝑇), if 𝑤𝑠 ≥ 2𝑤𝑁 .
EXAMPLE OF SINUSOIDAL SIGNAL
RECONSTRUCTION OF A BANDLIMITED SIGNAL

The reconstruction is given
by
𝑥𝑟 𝑡
∞
=
𝑛=−∞
sin(𝜋 𝑡 − 𝑛𝑇 /𝑇)
𝑥(𝑛)
𝜋 𝑡 − 𝑛𝑇 /𝑇
INTUITIVE EXPLANATION
It can be used for D/C
converter:
𝑋𝑟 𝑗𝑤 = 𝐻𝑟 𝑗𝑤 𝑋(𝑗𝑤)

DISCRETE-TIME PROCESSING

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

We can use a discrete-time low pass filter (LPF)
to do the low pass filtering for continuous time
signal.
EXAMPLE: LTI AND LPF

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

We can also use continuous-time system to
process discrete-time signals.
RESAMPLING: DOWNSAMPLING

The downsampling 𝑥𝑑 𝑛 = 𝑥(𝑛𝑀) implies
1
𝑋𝑑 𝑗𝑤 =
𝑀
𝑀−1
𝑖=0
𝑤
𝑋(𝑗( − 2𝜋𝑖/𝑀))
𝑀
INTUITION IN THE FREQUENCY DOMAIN
With
aliasing
Without
aliasing
DECIMATOR

A general system for downsampling by a factor
of M is the one shown above, which is called a
decimator.
UPSAMPLING

The upsampling is given by 𝑥𝑖 𝑛 = 𝑥(𝑛/𝐿),
where L is the integer factor.
EXPANDER
The output of
expander is given
by 𝑥𝑒 𝑛
= ∞
−∞ 𝑥(𝑘)𝛿(𝑛 − 𝑘𝐿
.
 In the frequency,
we have 𝑋𝑒 𝑤
= 𝑋 𝑤𝐿 .

INTERPOLATOR

It can be shown that the above structure realizes the
upsampling and interpolates the signals between samples:
∞
sin(𝜋(𝑛 − 𝑘𝐿)/𝐿)
𝑥𝑖 𝑛 =
𝑥(𝑘)
𝜋(𝑛 − 𝑘𝐿)/𝐿
−∞
SIMPLE AND PRACTICAL INTERPOLATION

The ideal interpolator is impossible to implement.
In practice, we can use a linear interpolator:
𝑛
ℎ 𝑛 = 1 − 𝐿 , |𝑛| ≤ 𝐿
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
TIME AND FREQUENCY OF LINEAR
INTERPOLATOR
CHANGING SAMPLING RATE BY A NON-INTEGER
FACTOR

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

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

The operations of linear filtering and
downsampling / upsampling can be exchanged
if we modify the linear filter.
MULTISTAGE DECIMATION


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

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

We can use oversampled A/D to simplify the
continuous-time antialiasing filter.
FREQUENCY DOMAIN INTUITION

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 < 𝑡 < 𝑇
ℎ0 𝑡 =
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

QUANTIZATION
This quantizer
is suitable for
bipolar signals.
 Generally, the
number of
quantization
levels should be
a power of tow,
but the number
is usually much
larger than 8.

ILLUSTATION
D/A CONVERSION

The ideal D/A is given by
∞
sin(𝜋 𝑡 − 𝑛𝑇 /𝑇)
𝑥𝑟 𝑡 =
𝑥(𝑛)
𝜋 𝑡 − 𝑛𝑇 /𝑇
𝑛=−∞
In practice, we need to use the above structure.
OVERSAMPLING


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.