Digital Signal Processing Fall 2009
Transcript of Digital Signal Processing Fall 2009
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Digital Signal Processing
Fall 2009
Lecture 2Fourier Transform and
Frequency Response
Book Reading for Lecture 1 & 2
Oppenheim: Pages 1-70
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Course at a Glance
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In Todays Lecture
Continuous and Discrete Frequency
Convolution Sum
Properties of LTI Systems Linear constant-coefficient difference
equations
Fourier Transform and Frequency Response
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Continuous Sinusoid
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Continuous Sinusoid (Cont..)
For every fixed value of the frequency F x(t) is
periodic.
T=1/F is the fundamental period of the sinusoidal
signal.
Continuous-time sinusoidal signals with distinct
frequencies are themselves distinct.
Increasing the frequency F results in an increase inthe rate of oscillation of the signal, in the sense that
more periods are included in a given time interval.
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Discrete Sinusoid
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Discrete Sinusoid (Cont..)
is the frequency in radians (since n is
dimensionless)
For close analogy with continuous time wespecify the units of radians/sample and the
units of n to be samples.
If we define = 2f then frequency f hasdimensions of cycles/sample.
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Relationship
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Periodicity of Discrete Sinusoid
The smallest value ofN for which above equation istrue is called the fundamental period of the sinusoid.
For this to be true there must exist an integerk suchthat
Thus a discrete time sinusoid is periodic only if itsfrequency is a rational number
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Aliased Frequencies
Discrete time sinusoids whose frequencies are
separated by an integer multiple of 2 are identical.
Thus all sinusoidal frequencies k are
indistinguishable. Where,
Any sinusoidal with an angular frequency that fallsoutside the interval to is identical to sinusoidal
frequency that falls within the fundamental interval.
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Convolution Sum
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Forming the sequence h[n-k] (Fig 2.9 Page 25)
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Computation of the Convolution Sum
Obtain the sequence h[n-k]
Reflecting h[k] about the origin to get h[-k]
Shifting the origin of the reflected sequence tok=n
Multiply x[k] and h[n-k] for inf < k < inf
Sum the products to compute the outputsample y[n]
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Computing a discrete convolution
Example 2.13 page 26
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Properties of LTI System
Commutative Property
Distributive Property
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Properties of LTI System (Cont..)
Cascaded System
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Properties of LTI System (Cont..)
Parallel System
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FIR Systems reflected in the h[n]
Finite-duration Impulse Response System
The impulse response has only a finite
number of nonzero samples. E.g. Ideal delay
Forward Difference
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IIR Systems - reflected in the h[n]
Infinite-duration Impulse Response System
The impulse response is infinitive in duration e.g.
Accumulator
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Stability
Another definition for stability is that a system
is stable if its impulse response is absolutely
sum able i.e.
FIR system are always stable, if each of h[n]values is finite in magnitude
IIR Systems may or may not be stable.
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Causality
A system is causal if h[n]=0 for n
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Inverse System
If
Then hi[n] is called inverse of h[n]
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LCCD equations
An important class of LTI systems: input and
output satisfy an Nth-order LCCD equations
Difference equation representation of the
accumulator
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Fourier Transform
Fourier transforms and frequency response
Frequency-domain representation of discrete-time
signals and systems
Symmetry properties of the Fourier transform
Fourier transform theorems
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Signal Representation
A sum of scaled, delayed impulse
Sinusoidal and complex exponential sequences
Sinusoidal input sinusoidal response with the same frequencyand with amplitude and phase determined by the system
Complex exponential sequences are eigenfunctions of LTI
systems.
signal representation based on sinusoids or complexexponentials
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Eigenfunctions
Complex exponentials as input to system h[n]
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Eigenvalue called frequency response
Frequency response is generally complex
describes changes in magnitude and phase.
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Frequency Response of the Ideal Delay
Example 2.17 page 41
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Frequency Response
The frequency response of discrete-time LTIsystems is always a periodic function of thefrequency variable with period 2.
Only specify over the interval < <
The low frequencies are close to 0.
The high frequencies are close to .
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Ideal frequency selective filters (Example 2.19)
For which the frequency response is unity
over a certain range of frequencies, and is
zero at the remaining frequencies.
Ideal low-pass filter: passes only low and rejects
high
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Sinusoidal response of LTI System
Example 2.18 page 42
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Signal Representation
More than sinusoids, a broad class of signals can berepresented as a linear combination of complexexponentials:
If x[n] can be represented as a superposition ofcomplex exponentials, output y[n] can be computedby using the frequency response, which is similar tothe function of impulse response.
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Frequency-domain representation of x[n]
By Fourier Transform Fourier Representation
These two equations together form a Fourierrepresentation for the sequence.
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In general Fourier transform is complex
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Frequency and impulse responses
Are a Fourier transform pair
Fourier transform is periodic with period 2
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Sufficient condition for Fourier transform
Condition for the convergence of the infinite sum
x[n] is absolutely summable, then its Fouriertransform exists (sufficient condition).
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Example: ideal low pass filter(Example 2.22 page 52)
Frequency response
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Fourier transform of a constant (Example 2.23 page 53)
Constant sequence
x[n]=1 for all n
Its not absolutely summable Its Fourier transform is defined as the
periodic impulse train
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Exercise 2
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Fourier Transform Properties
Makeup class timing??