digital signal processing fall 2009

8/14/2019 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



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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



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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??

digital signal processing fall 2009

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