Part 5 Response of Linear Systems

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Part 5Response of Linear Systems

• Linear Filtering of a Random Signals
• Power Spectrum Analysis
• Linear Estimation and Prediction Filters
• Mean-Square Estimation

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6. Linear Filtering of a Random Signal

• Linear System
• Our goal is to study the output process
  statistics in terms of the input process
  statistics and the system function.

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Deterministic System
Deterministic Systems
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Memoryless Systems
The output Y(t) in this case depends only on the
present value of the input X(t). i.e.,
.
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Linear Time-Invariant Systems
Time-Invariant System Shift in the input results
in the same shift in the output. Linear
Time-Invariant System A linear system with
time-invariant property.
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Linear Filtering of a Random Signal
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Theorem 6.1
Pf
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Theorem 6.2

• If the input to an LTI filter with impulse
  response h(t) is a
• wide sense stationary process X(t), the output
  Y(t) has the
• following properties
• Y(t) is a WSS process with expected value
• autocorrelation function
• (b) X(t) and Y(t) are jointly WSS and have I/O
  cross- correlation by
• (c) The output autocorrelation is related to the
  I/O
• cross-correlation by

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Theorem 6.2 (Contd)
Pf
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Example 6.1
X(t), a WSS stochastic process with expected
value ?X 10 volts, is the input to an LTI
filter with What is the expected value of the
filter output process Y(t) ? Sol
Ans 2(e0.5?1) V
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Example 6.2
A white Gaussian noise process X(t) with
autocorrelation function RW (? ) ?0? (? ) is
passed through the moving- average filter For
the output Y(t), find the expected value EY(t),
the I/O cross-correlation RWY (? ) and the
autocorrelation RY (? ). Sol
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Theorem 6.3
If a stationary Gaussian process X(t) is the
input to an LTI Filter h(t) , the output Y(t) is
a stationary Gaussian process with expected
value and autocorrelation given by Theorem
6.2. PfOmit it.
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Example 6.3
For the white noise moving-average process Y(t)
in Example 6.2, let ?0 10?15 W/Hz and T
10?3 s. For an arbitrary time t0, find PY(t0) gt
4?10?6. Sol
Ans Q(4) 3.17?10?5
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Theorem 6.4
The random sequence Xn is obtained by sampling
the continuous-time process X(t) at a rate of
1/Ts samples per second. If X(t) is a WSS
process with expected value EX(t) ?X and
autocorrelation RX (? ), then Xn is a WSS random
sequence with expected value EXn ?X and
autocorrelation function RX k RX (kTs). Pf
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Example 6.4
Continuing Example 6.3, the random sequence Yn is
obtained by sampling the white noise
moving-average process Y(t) at a rate of fs
104 samples per second. Derive the
autocorrelation function RY n of Yn. Sol
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Theorem 6.5
If the input to a discrete-time LTI filter with
impulse response hn is a WSS random sequence,
Xn, the output Yn has the following properties.
(a) Yn is a WSS random sequence with expected
value and
autocorrelation function (b) Yn and Xn
are jointly WSS with I/O cross-correlation (c)
The output autocorrelation is related to the I/O
cross- correlation by
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Example 6.5
A WSS random sequence, Xn, with ?X 1 and
auto- correlation function RXn is the input to
the order M?1 discrete-time moving-average
filter hn where For the case M 2, find the
following properties of the output random
sequence Yn the expected value ?Y, the
autocorrelation RYn, and the variance
VarYn. Sol
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Example 6.6
A WSS random sequence, Xn, with ?X 0 and
auto- correlation function RXn ? 2?n is
passed through the order M?1 discrete-time
moving-average filter hn where Find the output
autocorrelation RYn. Sol
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Example 6.7
A first-order discrete-time integrator with WSS
input sequence Xn has output Yn Xn 0.8Yn-1 .
What is the impulse response hn ? Sol
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Example 6.8
Continuing Example 6.7, suppose the WSS input Xn
with expected value ?X 0 and autocorrelation
function is the input to the first-order
integrator hn . Find the second moment, EYn2 ,
of the output. Sol
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Theorem 6.6
If Xn is a WSS process with expected value ? and
auto- correlation function RXk, then the vector
has correlation matrix and expected value
given by
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Example 6.9
The WSS sequence Xn has autocorrelation function
RXn as given in Example 6.5. Find the
correlation matrix of Sol
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Example 6.10
The order M?1 averaging filter hn given in
Example 6.6 can be represented by the M element
vector The input is The output vector
, then .

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• Linear Filtering of a Random Signals
• Power Spectrum Analysis
• Linear Estimation and Prediction Filters
• Mean-Square Estimation

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7. Power Spectrum Analysis

• Definition Fourier Transform
• Definition Power Spectral Density

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Theorem 7.1
Pf
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Theorem 7.2
Pf
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Example 7.1
Sol
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Example 7.2
A white Gaussian noise process X(t) with
autocorrelation function RW (? ) ?0? (? ) is
passed through the moving- average filter For
the output Y(t), find the power spectral density
SY (f ). Sol
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Discrete-Time Fourier Transform (DTFT)
Definition
Example 7.3 Calculate the DTFT H(?) of the
order M?1 moving-average filter hn of Example
6.6. Sol
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Power Spectral Density of a Random Sequence
Definition
Theorem 7.3 Discrete-Time Winer-Khintchine
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Theorem 7.4
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Example 7.4
Sol
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Example 7.5
Sol
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Cross Spectral Density
Definition
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Example 7.6
Sol
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Example 7.7
Sol
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Frequency Domain Filter Relationships
x(t)
Time DomainY(t) X(t)?h(t) Frequency Domain
W(f) V(f)H(f)
where V(f) FX(t),
W(f) FY(t), and
H(f) Fh(t).
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Theorem 7.5
Pf
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Example 7.8
Sol
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Example 7.9
Sol
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Example 7.10
Sol
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Theorem 7.6
Pf
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I/O Correlation and Spectral Density Functions
Time Domain
Frequency Domain
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• Linear Filtering of a Random Signals
• Power Spectrum Analysis
• Linear Estimation and Prediction Filters
• Mean-Square Estimation

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8. Linear Estimation and Prediction Filters

• Linear Predictor
• Used in cellular telephones as part of a speech
  compression algorithm.
• A speech waveform is considered to be a sample
  function of WSS process X(t).
• The waveform is sampled with rate 8000
  samples/sec to produce the random sequence Xn
  X(nT).
• The prediction problem is to estimate a future
  speech sample, Xnk using N previous samples
  Xn-M1 , Xn-M2 , , Xn.
• Need to minimize the cost, complexity, and power
  consumption of the predictor.

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Linear Prediction Filters
Use to estimate a future sample XXnk. We
wish to construct an LTI FIR filter hn with input
Xn such that the desired filter output at time n,
is the linear minimum mean square error
estimate Then we have The predictor can be
implemented by choosing .
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Theorem 8.1
Let Xn be a WSS random process with expected
value EXn 0 and autocorrelation function
RXk. The minimum mean square error linear
filter of order M?1, for predicting Xnk at time
n is the filter such that where is
called as the cross-correlation matrix.
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Example 8.1
Xn be a WSS random sequence with EXn 0 and
autocorrelation function RXk (? 0.9)k. For
M 2 samples, find , the
coefficients of the optimum linear predictor for
X Xn1, given . What is
the optimum linear predictor of Xn1 , given Xn
?1 and Xn. What is the mean square error of the
optimal predictor? Sol
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Theorem 8.2
If the random sequence Xn has a autocorrelation
function RXn bk RX0, the optimum linear
predictor of Xnk, given the M previous samples
is and the
minimum mean square error is
. Pf
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Linear Estimation Filters
Estimate XXn based on the noisy observations
YnXnWn. We use the vector
of the M most recent observations. Our
estimates will be the output resulting from
passing the sequence Yn through the LTI FIR
filter hn. Xn and Wn are assumed independent WSS
with EXnEWn0 and autocorrelation function
RXn and RWn. The linear minimum mean square
error estimate of X given the observation Yn is
Vector From
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Theorem 8.3
Let Xn and Wn be independent WSS random processes
with EXnEWn0 and autocorrelation function
RXk and RWk. Let YnXnWn. The minimum mean
square error linear estimation filter of Xn of
order M?1 given the input Yn is given by such
that
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Example 8.2
The independent random sequences Xn and Wn have
expected zero value and autocorrelation function
RXk (?0.9)k and RWk (0.2)?k. Use M 2
samples of the noisy observation sequence Yn
XnWn to estimate Xn. Find the
linear minimum mean square error prediction
filter Sol
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• Linear Filtering of a Random Signals
• Power Spectrum Analysis
• Linear Estimation and Prediction Filters
• Mean-Square Estimation

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9. Mean Square Estimation
Linear Estimation Observe a sample function of a
WSS random process Y(t) and design a linear
filter to estimate a sample function of another
WSS process X(t), where Y(t) X(t)
N(t). Wiener Filter The linear filter that
minimizes the mean square error. Mean Square
Error
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Theorem 9.1Linear Estimation
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Example 9.1
Sol