digital signal processing lectures

8/10/2019 Digital Signal Processing Lectures

1/39

Digital Signal Processing

Soma Biswas

Department of Electrical Engineering

IISc, Bangalore

Office: 215 A (EE)E-mail: [email protected]

Class Hours: Tuesday and Thursd ay 5:15 to 6.40 pm

Class Room: EE 218


2/39


3/39

19/09/2013 Digital Signal Processing - Lecture 9 3

The Fourier series representation of a continuous-time periodic signalgenerally requires infinitely many harmonically related complexexponentials.

Whereas the Fourier series of any discrete-time signal with period

requires only harmonically related complex exponentials. Because of the inherent periodicity of the complex exponentials with

period .

is an integer. Consequently, the set of periodic complex exponentials , , , defines all the distinct periodic complex exponentialswith frequencies that are integer multiples of (/).

Thus, the Fourier series representation of a periodic sequence []

contains only of these complex exponentials, and hence, it has a form


4/39


The Fourier series coefficients []are obtained by:

Note that the sequence []is periodic with period N: i.e.,

= , = [ + ], and more generally,


5/39


The Fourier series coefficients []can be interpreted to be a sequence offinite length, for = , , , ( ), and zero otherwise, or as a periodicsequence defined for all .

Both of these interpretation are acceptable. Advantage of interpreting the

Fourier series coefficients as a periodic sequence is that there is a dualitybetween the time and frequency domains for the Fourier seriesrepresentation of periodic sequences.

For convenience in notation, we use = (/).

With this notation, the discrete Fourier series (DFS) analysis-synthesis pairis expressed as follows:


6/39

Example


Representation of periodic impulse train in terms of sum of complex exponentials.

Complex exponentials with same magnitude and phase adds to unity at integer multiples ofand zero for all other integer.


7/39


8/39


9/39

Properties of DFS


Duality:

Periodic convolution:


10/3919/09/2013 Digital Signal Processing - Lecture 9 10


13/39

Fourier Transform of periodic signals


Uniform convergence of Fourier transform of a sequence requires thesequence to be absolutely summable, and mean-square convergencerequires that the sequence be square summable.

Periodic sequences satisfy neither condition, because they do notapproach zero as approaches .

However, sequences that can be expressed as a sum of complexexponentials can be considered to have a Fourier transform representation.

It is useful to incorporate the discrete Fourier series representation of

periodic signals within the framework of the Fourier transform. This can be done by interpreting the Fourier transform of a periodic signal

to be an impulse trans in the frequency domain within the impulse valuesproportional to the DFS coefficients for the sequence.

Specifically, if []is periodic with period and the correspondingdiscrete Fourier series coefficients are [], then the Fourier transform of

[]is defined to be the impulse train



Inverse Fourier transform

Fourier transform, DTFT


15/39

Although the Fourier transform of periodic sequence does not converge inthe normal sense, the introduction of impulses permits us t includeperiodic sequences formally within the framework of Fourier transformanalysis.

This approach can be used to obtain a Fourier transform representation ofother nonsummable sequences.



17/39

In other words, the periodic sequence []of DFS coefficients has aninterpretation as equally spaced samples of the Fourier transform of thefinite-length sequence obtained by extracting one period of [].

This corresponds to sampling the Fourier transform at equally spacedfrequencies between = and = with frequency spacing of /.



18/39


19/39

24/09/2013 19Digital Signal Processing - Lecture 10


20/39

Digital Signal Processing - Lecture 10

Example

24/09/2013 20


21/39



22/39

Properties of DFT

24/09/2013 22

Linearity:

Clearly if []has length and 2[]has length 2, then the maximum length of3[]will be 3= max[, 2].

Thus both the DFTs must be computed with the same length > 3. Zero padding is done to match length.

Circular shift:



23/39



24/39

24/09/2013 24

Duality:



25/39

Properties of DFT

24/09/2013 25

Symmetry:

Circular convolution:



26/39


Circular convolution of finite length

Sequence with single delayed impulse


27/39

Example



28/39

Example



29/39

24/09/2013Digital Signal Processing - Lecture

10 29


30/39

Linear convolution using DFT

24/09/2013 30

Efficient algorithms are available for computing the discrete Fouriertransform of finite-duration sequence. They are known as fastFour ier Trans form(FFT) algorithms.

Because of these algorithms are available, it is computationallyefficient to implement a convolution of two sequences by thefollowing procedure:

Compute N-point DFTs()and()of the two sequences[]and [], respectively.

Compute the product = .

Compute the sequence = []as the inverse DFT of .

This will result in circular convolution, however, in most cases wewant to implement linear convolution of two sequences, i.e., in caseof LTI systems.

To obtain linear convolution, we must ensure that the circularconvolution has the effect of linear convolution.



31/39

Linear convolution of two finite-length sequences


Therefore, ( + 1)is the

maximum length of the sequence

3[]resulting from the linear

convolution of a sequence of length with a sequence of length .


32/39


10 32

Time aliasing in the circular

convolution of two finite length

sequences can be avoided if + 1, Also it is clear that if = =

, all of the sequence values of the

circular convolution may be different

from those of the linear convolution.


33/39

Contd.



34/39

Contd.



35/39

Contd.


I l ti LTI t i DFT


36/39

Implementing LTI system using DFT

24/09/2013 36

Since LTI systems can be implemented by convolution, circularconvolution can be used to implement these systems.

Lets us first consider an L-point input sequence []and a P-pointimpulse response []. The linear convolution of these twosequences, which will be denoted by [], has finite duration length( + ).

If a circular convolution is done with at least ( + )points, itwill be identical to linear convolution.

The circular convolution can be achieved by multiplying the DFTsof []and []. Both []and []must be augmented withsequence values of zero amplitude. This process is often referred toas zero-padding.

The output of a FIR system whose input also has finite length can

be computed with DFT. In many applications, such as filtering of speech waveform, the

input signal is of indefinite duration. While, theoretically, we mightbe able to store the entire waveform and then implement theprocedure using DFT for a large number of points, however, suchDFT is generally impractical to compute.



37/39


10 37

Another drawback is that in this method, no filtered output samples

can be computed until all the input samples have been collected.Generally we would like to avoid such delay in processing.

The solution to both problems is b lock con vo lu t ion, in which thesignal to be filtered is segmented into sections of length .

Each section can then be convolved with the finite-length impulse

response and the filtered sections fitted together in an appropriateway.

The linear filtering of each block can then be implemented using theDFT.


38/39


10 38

We assume = 0 < 0and

the length of []is much greater

than .

The sequence []can be

represented as a sum of shifted finite-length segments of length ; i.e.,

Convolution Is LTI operation


39/39

Over lap-save method

digital signal processing lectures

Documents