206554: Digital Signal Processing

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20-6554 Digital Signal Processing

• Chapter 1
• Introduction
• (pp1-31 of Lynn Fuerst)

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Discrete-Time Signals
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Signal denoted by xn --- e.g. x0, x1, x2,
, xk,
Sampling interval T
Convention input denoted by xn, output by yn
Practical Applications Smooth out rapid
fluctuations by using a running-mean this is a
digital low-pass filter
(non-recursive)
In recursive form
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Sampling, and A-D conversion
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Equally spaced samples, interval T. How often
do we need to sample to fully represent a given
signal?
Shannon's Sampling Theorem "An analog signal
containing components up to some maximum
frequency f1 Hz may be completely represented by
regularly spaced samples, provided the sampling
rate is at least 2f1 samples per second
Note - this is two samples per period of the
highest frequency present.
Note - the sampling interval is then
If the sampling interval is T1 then the highest
frequency which can be represented is
or alternatively
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Jargon
Nyquist frequency Maximum frequency f1 contained
in the analog signal.
Nyquist rate Minimum sampling rate (2f1 samples
per second) at which the signal can be recovered
Folding frequency Half the sampling frequency,
the highest frequency which can be represented.
The spectrum of a sampled signal repeats around
multiples of the sampling frequency. If the
signal contains frequencies higher than the
folding frequency, then repetitions of the
spectrum overlap and cause distortion when you
try to reconstruct the signal. This is known as
ALIASING.
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Basic Types of Digital Signal
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Unit step
Unit impulse
Unit ramp
Note that
and
and that
and
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Periodic signals
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Two main problems 1 If we are to accurately
represent a periodic signal, we must sample it in
such as way that some integer multiple of the
sampling interval is equal to the period of the
signal. This can be expressed in the following
way
where the pattern repeats every N sample
intervals and m is an integer. ? is the number
of radians per sampling interval.
2 A second problem concerns the frequency scale.
As indicated above, there are 2?/? samples per
period (dimensionless frequency scale). If we do
need to use a frequency scale, we can use the
actual sampling interval T to give the sampling
frequency
The digital signal is then
.
and since nT represents time in seconds, ?
represents frequency in Hz, so
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Ambiguity in Digital Signals
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Inherent ambiguity, since any set of fixed points
can by connected in an infinite number of ways.
Sampled sinusoids can represent not just the
underlying signal from which they were derived,
but from sinusoids of higher frequencies
(multiples of 2?)
Met earlier Aliasing.
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Ambiguity in Digital Signals, contd
7a
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Digital Processors
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This course restricts itself to LTI (Linear
Time-Invariant) systems.
Linear system obeys the principle of
superposition -
if input
leads to output
(ai constants)
then input
leads to output
A consequence is the property of frequency
preservation (the output can only contain those
frequencies present in the input)
Time invariant means that system characteristics
do not change with time
i.e. Shifting the input shifts output by the same
amount.
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Other system properties
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Terms worth knowing about
Causality
Output signal depends only on present and/or
previous values of input.
Stability
Produces a finite or bounded response to a
bounded input.
Invertibility
If a system turns input xn into output yn,
then its inverse (if it exists) turns input yn
into output xn.
Memory
A system with memory can calculate its output at
step n using not only xn but also previous
values xn-1, xn-2 etc.
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Ch 1 Problems
Q1.4
Q1.6
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Q1.7(a)
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Q1.7(c)
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Q1.7(c)
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Q1.7(c)