206554: Digital Signal Processing

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20-6554 Digital Signal Processing

• Chapter 10
• Random DSP

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10.2
Chapter 10 Introduction
DSP principally concerned with processing
signals this chapter aims to answer the
following questions

• How are the statistical measures of random
  signals and noise affected by processing?

• If we know these at the input to a
  filter/process, can we predict what they will be
  at the output?

• Can we design an optimum processor for enhancing
  a signal contaminated by noise?

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10.3
10.2 Response of Linear Processors
Compare input and output of an LTI processor to a
random sequence. Reconsider digital convolution
of this signal. Section 2.4 illustrated the
process of using the impulse response of a
system, hn, to find the response to a signal
xn (flip and shift). Apply the same process
here.
For a random signal - do not know the individual
sample values, just the statistical properties -
cannot do the calculation directly. However

• Outputs are the weighted sum of random inputs,
  the weights from hn. Successive outputs are now
  not mutually independent.

• The correlation increases with the number of
  terms in hn a long hn implies a narrow
  frequency band. Narrow band-pass filters
  therefore result in highly correlated output data.

• If the input is white then the outputs are
  weighted sums of random numbers for which the
  means and variances are additive. The mean of
  the output is then the mean of the input
  multiplied by the sum of the impulse response
  terms.

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10.4
The mean expected value of the output sequence is
The mean of the output is the mean of the input
multiplied by the sum of the impulse response
terms. Intuitive remember the GAIN of the
filter? See the example 10.1 (p319) included
as sec10.1.xls.
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10.5
Output ACF
from the convolution sum, we can follow the maths
on pp320,321 to get
, where
is the ACFN of
Note that jq is not strictly speaking the ACF
of hn since it is not the average product.
In summary The ACF of the output of a LTI
processor is the convolution of the ACF of the
input with the ACFN of the impulse response of
the processor.
The Excel worksheet 10.2, in the workbook
sec10.1.xls illustrates this.
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10.6
Variance
Output variance central value of output
covariance function
and since
we get
For zero mean,
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10.7
Power Spectrum
Output power spectrum is the input spectrum
multiplied by the squared magnitude of the
processors frequency response
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10.8
10.3 White Noise Through a Filter
Must process a signal in an appropriate fashion
if it is noise, we may wish to filter it out if
it is a wanted signal we may need to enhance or
extract it. Measures such as the mean,
variance, ACF and power spectrum are used.
The previous formulae are simplified for white
noise, which has zero mean and unit variance.
The statistical properties of the input are
therefore
and equation 9.14
gives
confirming that the power spectrum of the
sequence is flat or white.
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10.9
After filtering, the mean value of the output,
will be 0.
For the ACF
so the ACF of the output the ACF of the
filter/processor itself
For the variance
So, for white noise input, the output properties
(ACF and spectrum) reflect the properties of the
processor rather than the signal.
The statistical properties of the output are
therefore
See worksheet 10.6 to illustrate Figures 10.6 and
10.7 from the book.
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10.10
10.4 System Identification by Cross-Correlation
For input in the form of white noise, the CCF
between the input and the output characterises
the system any non-whiteness in the output
is due to the frequency-selective properties of
the system itself.
The cross-correlation function is
The output yn is the convolution of the input
xn with the impulse response hn.
For white noise input (mean0, variance1), the
ACF is a delta function, so
So, given a white noise input, the input/output
CCF is identical to the systems impulse
response, which completely defines it in the time
domain.
See example on worksheet 10.8.
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Generate random xn and apply a filter hn by
convolution to get yn. Take the CCF of this x
and y and find that it matches hn
10.11
In the frequency-domain convolution is replaced
by multiplication
For white noise (mean0, variance1)
so
Note the non-linear discussion on p332/3.
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10.12
10.5 Signals in Noise
One of the most important application areas of
DSP is the extraction of signals from noise.
There are usually three aims

• Recovery extracting the detail of the signal
  from the noise. We will look at an example
  dealing with narrow-band signal with wide-band
  noise.

• Detection determining whether a signal of known
  shape is present. Dealt with using a technique
  known as matchedfiltering.

• Enhancement of a repetitive signal by averaging
  over many repetitions.

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10.13
1. Signal Recovery
Recovery of narrow-band signal from wide-band
noise. Predict improvement in signal/noise ratio
that can be gained from linear filtering.
Output noise power is
If the form of
is as given in fig 10.10
and if the noise spectrum is white,
So the reduction in total noise power (or
variance) through the filter is just the ratio of
the filter bandwidth to
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10.14
2. Matched-Filter Detection
One of the most important techniques. In this
case we assume we know the shape (or waveform) of
the signal we are looking for. We need to
determine if and when it occurs.
Convolution and correlation are very similar
the only real difference is that in the case of
convolution one of the two signals being
multiplied and summed is reversed in time.
The operation of the matched filter illustrates
this it is in effect a matched correlator,
producing an output similar in form to the ACF of
the signal to which it is matched. The matched
filter is the signal xn, but time-reversed.
The output from a matched filter is a signal, and
a function of the time parameter n, not the lag
m. Its peak central value occurs at the instant
when the complete signal has entered the filter,
not at n0, but the form is the same as the ACF.
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10.15
Object is not to extract a signal but to find a
filter that gives the maximum output whenever the
signal waveform occurs.
See example worksheet 10.13
Case 1
Matched filter behaviour when the data have noise
with 0 variance added
Case 2
The noise has a variance of 0.3. The signal is
still evident in the ACF. In both cases the
ACFs have been offset by 1.5 for visibility.
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10.16
To determine the improvement in signal to noise
ratio, for the special case of white noise
The peak input signal is denoted by
The peak output is the central value of the ACF
The variance of white noise through a linear
filter increases by a factor
so the overall S/N ratio due to matched filtering
(measured via signal and noise amplitudes) is
For the data used in the example above,
1
The S/N is calculated in the spreadsheet.
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10.17
Cannot improve the S/N by repeating the matched
filtering the output is already optimal.
The S/N improvement will increase for longer
signals, and for signals which are more spread
in time the earlier equation above shows that
this improvement depends upon the ratio between
the sum of the squares of the sample values and
the peak sample value.
The best results need a signal with a large sum
of squares (large total energy) and a small peak
value. The signal needs to be spread out with
values more or less equal. For the purposes of
detection it would be helpful for the signal to
have a strong peak zero ACF and small values
elsewhere. Such a signal is known as a
Pseudo-Random Binary Sequence (PRBS) a random
series of values 1 and -1.
In worksheet 10.14 in the current Excel workbook
this is simulated, reproducing Figure 10.14
(p345) of the book.
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10.18
A PRBS of length 64 is generated, and applied
three times to a data set of length 420 points,
with two of them overlapping.
The matched filters impulse response (as normal)
is a time-reversed version of this, and the
result of the filter is obtain by convolving the
two. This is illustrated in the figure which
is presented for several cases no noise, noise
with a variance of 0.3 and (illustrating how well
this process works, with a variance of 1).
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10.19
Note some real-world examples of this on p346.
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10.20
10.5.4 Signal Averaging
For a repeated signal (over a known interval) in
the presence of noise, we can improve the S/N
ratio by averaging over a number of samples.
Simulated in worksheet 10.15, where a repeated
delta function has been filtered. Noise is
added with a given variance the output is
averaged a number of times. The principle is
that the signal values add but the noise values
average out.
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10.21
SUMMARY
Output values
Output mean
Output ACF
Output variance
when input is random and white,
,
,
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10.22
Summary 2 Power Spectra and ACF/CCF
)
(since
for white noise input,
and
for white noise input,
BSc 4 Mathematics. 20-6554 Digital Signal
Processing