Advanced Digital Signal
Processing -Introduction
LECTURE-I
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Discrete-Time Signals
The entire subject is focused on discrete time signals, which
usually are A/D converted signals
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Discrete-Time Signals
• An important signal is the unit sample
• It can be used to represent other discrete time signals
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Discrete-Time Signals
• The unit step
• Is related to the unit sample by
• A complex exponential is important for Fourier decomposition
of signals
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Discrete-Time Systems
Any system that maps an input sequence to an output sequence
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Discrete-Time Systems
• For a linear system, this must hold, where a and b are arbitrary
constants
• Note that this implies, that if the input is written as sum over unit
impulses
• the output is
• where
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Discrete-Time Systems
• For a shift-invariant system, if the input is shifted by n0 then
the output must also shift by n0. I.e., the system does not
change over time
• For a system that is both linear and shift invariant (LSI) the
following holds
• This is called the convolution sum, which is also written as
• Hence an LSI system is completely specified by its unit
sample response h(n)
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Discrete-Time Systems
• A system is called causal if at any time n0 the output of
the system only depends on input values for n<=n0
• An LSI system can only be causal if h(n)=0 for n<0
• A LSI system is called stable in the Bounded-Input
Bounded-Output sense if
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Time-Domain Description of LSI Filters
• An important class of linear shift-invariant systems are those
whose input x(n) and output y(n) are related by a linear
constant coefficient difference equation
• p and q determine the order of the system and a(k) and b(k)
are the filter coefficients. This is often written in this form
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Time-Domain Description of LSI Filters
• If p=0, then the following form result and hence the unit
sample response h(n) is finite in length, hence these filters are
called finite impulse response filter (FIR)
• If p is not zero then the impulse response is not finite hence
the filters are called IIR filters
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Discrete-Time Fourier Transform
• The discrete-time Fourier transform (DTFT) is defined as
• A sufficient condition that this sum converges is
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Discrete-Time Fourier Transform
• The frequency response of a LTI filter is the DTFT of the unit
sample response
• The DTFT is invertible
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Discrete-Time Fourier Transform
• The convolution theorem is quite helpful
• Another useful theorem is Parseval’s theorem
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Discrete-Time Fourier Transform
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The z-Transform
• A generalization to the DTFT, allowing many signals
without DTFT to be transformed
• Note that for z=ejw holds
• Note that the sum might not converge for all possible z,
hence there is always a region of convergence associated
with a z transform
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The z-Transform
• When the z-transform is used for power spectral densities,
then the following property comes in handy
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The z-Transform
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The z-Transform
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The z-Transform
• The z-transform of the unit sample response of a LTI
system will become important later on
• For
FIR
filters
this
becomes
and the roots of this polynomial are called the zeros of the
filter
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The z-Transform
• For IIR filters this becomes and the roots of the denominator
polynomial are called the poles of the filter
• If the coefficients are real-valued (equivalently h(n) is real
valued) then and all poles and zeros will occur in conjugate
pairs
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The z-Transform
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Special Classes of Filters
• Generalized linear phase filters have the following frequency
response, where A(ejw) is a real valued function and a and b
are constants
• This can be achieved with FIR filters having either conjugate
symmetric (Hermitian) or conjugate anti symmetric (antiHermitian) unit sample responses
• This though implies in other words, if H(z) has a zero at z0
then H(z) must also have a zero at 1/z0*
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Special Classes of Filters
• All pass filters are another special class of filters, which
have a frequency response with constant magnitude
• An all pass filter having a rational system function H(z)
must be of the following form
• Hence if H(z) has a pole at ak, then H(z) also has a pole at
1/ak*
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