2' Sinusoids

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• 1. ??
• 2. Sinusoids
• 3. Spectrum response
• 4. Sampling Aliasing
• 5. FIR filters
• 6. Frequency response of FIR filters
• 7. Z-transform
• 8. IIR filters
• 13. Computing the spectrum
• 10. ADSP 2181 Evaluation Board ??

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Analysis in different domains

• n -domain or time domain
• the domain of sequences, impulse responses and
  difference equations
• the real or actual domain where the signals
  are generated and processed,and where the
  implementation of filters takes place
• W domain or frequency domain
• the domain of frequency responses and spectrum
  representations
• physical significance when analyzing sound, but
  is seldom used for implementation
• z -domain
• the domain of z -transforms, operators, and poles
  and zeros
• primarily for its convenience in mathematical
  analysis and synthesis

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Comparison
Fourier transform Fourier series Laplace
transform z- transform
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Definition of the z -Transform
A finite-length signal xn
the z -transform of such a signal is defined by
z any complex number
z transform
inverse z transform
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Example 7-1. The z transform of a signal
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Example 7-2Inverse z -Transform
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The z -Transform and Linear Systems
The general difference equation of an FIR filter
Let the input to system be the signal
where z is any complex number
The corresponding output signal is
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System function
System function z -transform of the
impulse-response sequence
For FIR filters, if the input is zn for - ltnlt
, then the corresponding output is
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Example 7-3 Zeros of System Function
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Exercise 7-1
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Exercise 7-2
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The Superposition Property of the z -Transform
The z -transform is a linear transformation.
by considering the sequence xnax 1nbx 2n
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Example 7-4 z-transform of a Signal
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The Time-Delay Property of the z -Transform
The quantity z -1 in the z -domain corresponds to
a time shift of 1 in the n -domain.
Each of the signal samples has moved over one
position in the tablei.e.,ynxn-1
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unit-delay property
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The z -Transform as an Operator
Unit-Delay Operator
In the time domain, the unit-delay operator D is
defined by
The input to the unit-delay system be the signal
The output of the unit delay is
The unit-delay system is represented by the
operator z -1
cf)
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Operator Notation
The first difference of two successive signal
values
operator equation
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Operator Notation in Block Diagrams
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Convolution and the z -Transform
discrete convolution of two finite-length
sequences xn and hn is given by the formula
the z -transform of yn
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Example 7-5 Convolution via H(z)X(z)
The z -transform method can be used to convolve
the following signals
The z -transforms of the sequences xnand hnare
Both X(z)and H(z)are polynomials in z -1 ,so we
can compute the z -transform of the convolution
by multiplying these two polynomials
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Since the coefficients of any z -polynomial are
just the sequence values,with their position in
the sequence being indicated by the power of (z
-1 ), we can inverse transform Y(z) to obtain
Now we look at the convolution sum for computing
the output. If we write out a few terms,we can
detect a pattern that is similar to the z
-transform polynomial multiplication.
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Notice how the index of hk and the index of
xn-k sum to the same value (i.e.,n ) for all
products that contribute to yn. The same thing
happens in polynomial multiplication because
exponents add.
In Section 5-3.3.1 on p.110 we demonstrated a
synthetic multiplication tableau for evaluating
the convolution of xnwith hn. Now we see that
this is also a process for multiplying the
polynomials X(z) and H(z).The procedure is
repeated below for the numerical example of this
section.
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In the z -transforms X(z), H(z), and Y(z), the
power of z -1 is implied by the horizontal
position of the coefficient in the tableau. Each
row is produced by multiplying the xn row by
one of the hnvalues and shifting the result
right by the implied power of z -1 . The final
answer is obtained by summing down the
columns.The final row is the sequence of values
of ynxnhn or, equivalently, the
coefficients of the polynomial Y(z).
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Cascading Systems
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Exercise 7-6
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Factoring z -Polynomials
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Deconvolution
Can we use the second filter in a cascade to undo
the effect of the first filter? What we would
like is for the output of the second filter to be
equal to the input to the first.
Since the first system processes the input via
convolution, the second filter tries to undo
convolution, so the process is called
deconvolution .
Another term for this is inverse filtering , and
if H1(z) H2 (z)1, then H2(z) is said to be the
inverse of H1 (z)(and vice versa).
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This means that the inverse filter for an FIR
filter cannot be also an FIR filter.
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Relationship Between the z -Domain and the W
-Domain
Frequency response H(e j? ) is obtained from the
system function H(z) by evaluating H(z) for a
specific set of values of z.
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The z -Plane and the Unit Circle
The frequency response is periodic with period 2
p , we need only evaluate it over one period,
such as -plt?ltp.
If we substitute these values of ? into (7.26),
we see that the corresponding values of z all
have unit magnitude and that the angle ? varies
from -p to p .
The contour on which all the values of ze j?
lie is called the unit circle .
z -plan
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The Zeros and Poles of H(z)
FIR system is essentially determined by its zeros.
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Pole-zero plot
Each factor of the form (1 -az -1 ) represents a
zero at za and a pole at z0.
The unit circle is where H(z) is evaluated to
obtain the frequency response of the LTI system
whose system function is H(z).
pole-zero plot
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Significance of the Zeros of H(z)
Signals are of the form xnz0 n for all n
,where the subscript signifies that z 0 is a
particular complex number. The output is
If z 0 is one of the zeros of H(z), then H(z 0
)0 so the output will be zero.
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Nulling Filters
We have just shown that if the zeros of H(z) lie
on the unit circle, then certain sinusoidal input
signals are removed or nulled by the filter.
Zeros in the z -plane can remove only signals
that have the special form xnz0n. If we want
to eliminate a sinusoidal input signal, then we
actually have to remove two signals of the form
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Each complex exponential can be removed with a
first-order FIR filter, and then the two filters
would be cascaded to form the second-order
nulling filter that removes the cosine. The
second-order FIR filter will have two zeros at z
1e j? 0 and z 2e -j? 0. The signal z1n will
be nulled by a filter with system function
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Similarly,H2(z)1-z2z -1 will remove z 2n.
The nulling filter that will remove the signal
cos (0 .25 pn) from its input.
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Graphical Relation Between z and W
11-point running sum
the 10 zeros of H(z) at ze j 2 pk/11 , for k1
,2 ,...,10 and the 10 poles at z0
The frequency response is obtained by evaluating
the values of the z -transform along the unit
circle in Fig.7-7.
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Demo
FIR filter with two zeros. Notice changes in the
impulse response hn and the frequency response
as the complex zero pair is moved around the unit
circle (changing the angle of the zeros).
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FIR filter with three zeros one is held fixed at
z -1. Notice changes in the impulse response
hn and the frequency response as the complex
zero pair is moved around the unit circle
(changing angle).
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FIR filter with ten zeros equally spaced around
the unit circle. Notice changes in the impulse
response hnand the frequency response as the
zero at z 1, is moved radially
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FIR filter with ten zeros equally spaced around
the unit circle. Notice changes in the impulse
response hn and the frequency response as the
zero pair at 72 degrees is moved radially.
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The L -Point Running-Sum Filter
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A Complex Bandpass Filter
How to control the frequency response of an FIR
filter by placing its zeros on the unit circle.
the index k0 denotes the one omitted root at
zej2pk0 /L
k02
The resulting filter coefficients will be
complex-valued, if we realize that the zeros
cannot all be grouped as complex-conjugate pairs.
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H(z)
G(z)
the filter coefficients of the complex bandpass
filter
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A Bandpass Filter with Real Coefficients
the kth filter coefficient
We see that the two zeros at ze j 4 p/10e j
2p(2 )/10 and ze -j 4 p/10e j 2 p(8 )/10 have
been replaced by a single real zero.
The location of this new zero appears to be at
zcos (2pk0 /L)cos (0.4 p)0.309, which is the
real part of the missing unit-circle zeros.
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The exact location of the new zero
new zero,
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Practical Bandpass Filter Design
The z -transform converts difficult problems
involving convolution and frequency response
into simple algebraic ideas based on multiplying
and factoring polynomials.
For an FIR frequency selective (lowpass,
bandpass, highpass) filter, the width of the
transition region is inversely proportional to M
, the order of the system function polynomial.
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Properties of Linear-Phase Filters
The Linear-Phase Condition
FIR systems that have symmetric filter
coefficients ( symmetric impulse responses) have
frequency responses with linear phase.
system function
M4, L5
Substitute ,
General form,
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The frequency response of any symmetric filter
has the form of a real amplitude function R(ej?)
times a linear-phase factor e-j?M/2. The latter
factor corresponds to a delay of M/2 samples.
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Locations of the Zeros of FIR Linear-Phase Systems
If the filter coefficients satisfy the condition
bkbM-k , for k0 ,1 ,...,M ,
To demonstrate this reciprocal property of
linear-phase filters, consider a 4-point system
of the form,
z0
1/z0
1/z0
z0
The zeros that are not on the unit circle occur
are responsible for creating the passband of the
BPF. Zeros on the unit circle are mainly
responsible for creating the stopband of the
filter.
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