Mathworks SIGNAL PROCESSING TOOLBOX 6 Installation Guide

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Signal Processing

Getting Star ted Guide

Toolbox™ 6

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Signal Processing Toolbox™ Getting S tarted Guide

The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathW orks, Inc.

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Revision History

September 2006 First printing New for Version 6.6 (Release 2006b) March 2007 Online only Revised for Version 6.7 (Release 2007a) September 2007 Online only Revised for Version 6.8 (Release 2007b) March 2008 Online only Revised for Version 6.9 (Release 2008a) October 2008 Online only Revised for Version 6.10 (Release 2008b) March 2009 Online only Revised for Version 6.11 (Release 2009a) September 2009 Online only Revised for Version 6.12 (Release 2009b) March 2010 Online only Revised for Version 6.13 (Release 2010a)

Product Overvi

Product Summa Command-Line Graphical Us Supported Da

.................................

Functions and Objects

er Interfaces

ta Types

...........................

.............................

Overview

................

Contents

1-2 1-2 1-2 1-3 1-3

Central Fea

Signal Proc Signals and Filter Des Linear Sys Windowing Spectral Transfor Statisti Paramet Linear P Multira Wavefor Other O

active Tools

Inter

Exten

Find

sibility

Expan Expa

ing More Information

tures

essing Toolbox Functions

Systems

ign, Analysis, and Implementation

tem Transformations

Functions

Analysis

......................................

cal Signal Processing

ric Modeling rediction

te Signal Processing

mGeneration

perations

......................................

ding the Toolbox

ndingtheToolboxwithOtherToolboxes

..................................

..................

...............................

...........

.....................

..............................

..................................

........................

...............................

.................................

.........................

..............................

.................................

..................................

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...........

..........................

1-4 1-4 1-4 1-5 1-5 1-5 1-5 1-5 1-5 1-6 1-6 1-6 1-6 1-6

1-7

1-8 1-8 1-8

1-9

Basic Signal Processing Concepts

Representing Signals .............................. 2-2

Numeric Arrays Vector Represe n tatio n

................................... 2-2

............................. 2-2

Waveform Generation: Time Vectors and Sinusoids

Time Vectors Common Sequences: Unit Impulse, Unit Step, and Unit

Ramp Multichannel Signals Common Periodic Waveforms Common Aperiodic Waveforms The pulstran Function The Sinc Function The Dirichlet Function

Working with Data

Importing Data from Within the MATLAB E nvironment Importing Data from Outside the MATLAB

Environment Converting Data into a MAT-File Exporting Data Data Precision

Selected Bibliography

..................................... 2-4

......................................... 2-5

.............................. 2-5

........................ 2-6

...................... 2-7

............................. 2-8

................................. 2-9

............................. 2-10

................................. 2-12

................................... 2-12

.................... 2-12

................................... 2-13

.................................... 2-13

.............................. 2-14

... 2-4

.. 2-12

vi Contents

Filter Design with Fdesign and Filterbuilder

Filter Design Process Overview ..................... 3-2

Basic Filter Design Process

Using Filterbuild er to Design a Filter

......................... 3-4

............... 3-9

Filter Design with the FDATool GUI

Introduction ...................................... 4-2

Designing the Filter

Analyzing the Filter

Designing Additional Filters

Viewing and Annotating the Filter

Viewing the Filter in FVTool Using FVTool for Annotation

Exporting Filters from FDATool

Filtering with dfilt

Designing Filters using Comm a nd Line Functions

Where to Find More Information

............................... 4-3

............................... 4-8

........................ 4-10

.................. 4-11

........................ 4-11

........................ 4-15

.................... 4-17

................................. 4-18

.................... 4-24

Spectral Analysis

.... 4-21

Introduction ...................................... 5-2

Spectral Estimators Spectral Analysis Algorithms Spectral Analysis Objects

Creating a Spectral Analysis Object

Producing a PSD Estimate

Changing Spectral Analysis Object Property Values

............................... 5-2

........................ 5-2

........................... 5-3

................. 5-4

......................... 5-6

.. 5-8

vii

Using the set command to Set Property Values ......... 5-8

Using Options Objects to Set Property Values

.......... 5-11

Measuring Signal Power

........................... 5-13

Index

viii Contents

Overview

• “Product Overview” on page 1-2

• “Central Features” on p age 1-4

• “Interactive Tools” on page 1-7

• “Extensibility” on page 1-8

• “Finding More Information” on page 1-9

1 Overview

Product Overview

Product Summary

Signal Processing Toolbo x™ software is a collection of tools based on the MATLAB processing operations, from waveform generation to filter design and implementation, parametric modeling, and spectral analysis. The toolbox provides two categories of tools, command-line functions/objects and graphical user interfaces:

In this section...

“Product Summary” on page 1-2

“Command-Line Functions and Objects” on page 1-2

“Graphical User Interfaces” on page 1-3

“Supported Data Types” on page 1-3

environment. The toolbox supports a wide range of signal

1-2

Command-Line Functions and Objects

Command-line functions and objects are available in the following categories:

• Discrete-time filter design, analysis, and implementation

• Analog filter design, analysis, and implementation

• Linear system transformations

• Windowing functions

• Spectral analysis and cepstral analysis

• Transforms

• Statistical signal processing

• Parametric modeling

• Linear prediction

• Multirate signal processing

• Waveform generation

Product Overview

Note For information on objects, see Object-Oriented Programming.

Graphical User Interfaces

A suite of interactive graphical user interfaces are available for

• Filter design and analysis

• Window design and analysis

• Signal plotting and analysis, spectral analysis, and filtering

Supported Data Types

The Signal Processing Toolbox software supports only double-precision inputs. If you input single-precision floating-point or integer data, you should not expect to receive correct results, and in many cases, an error will occur. The Filter Design Toolbox™ product, i n conjunction with the Fixed-Point Toolbox™ product, enables single-precision floating-point and fixed-point support for filtering and filter design.

1-3

1 Overview

Central Features

In this section...

“Signal Processing T oolbox Functions” on page 1-4

“Signals and Systems” on page 1-4

“Filter Design, Analysis, and Implementation” on page 1-5

“Linear System Transformations” on page 1-5

“Windowing Functions” on page 1-5

“Spectral Analysis” on page 1-5

“Transforms” on page 1-5

“Statistical Signal Processing” on page 1-5

“Parametric Modeling” on page 1-6

“Linear Prediction” on page 1-6

“Multirate Signal Processing” on page 1-6

1-4

“Waveform Generation” on page 1-6

“Other Operations” on page 1-6

Signal Processing Toolbox Functions

Signal Processing Toolbox functions are algorithms that implement a variety of signal processing tasks. These toolbox functions are a specialized extension of the MATLAB computational and graphical environment.

Signals and Systems

The basic entities that toolbox functions work with are signals and systems. The functions emphasize digital (or discrete) signals and filters, as opposed to analog (or continuous) signals. The principal filter type the toolbox supports is the linear, time-invariant digital filter with a single input and a single output. You can represent linear time-invariant systems using one of several models (such as transfer function, state-space, zero-pole-gain, and seco n d-order section), and you can conv ert between representations.

Central Features

Filter Design, A

Signal Processi design. The majo IIR filter desi analog filter p

ng Toolbox software provides customizable support for filter

gn, analysis, and implementation, filter order estimation, and

rototyping and transformations.

Linear System

The toolbox h to and from se and transfer

as a number of transformation functions, including conversions

cond-order sections, state-space, pole-zero, lattice or ladder,

functions.

Windowing F

The toolbo graphical using thes

Spectral

Toolbox mean-sq nonpara in the to period comput one-si spect

xprovidesmanycommonlyusedwindowfunctionsaswellas

user interfaces to view and compare windows and design filters

ewindows.

Analysis

functions are available for estimating the power spectral density,

uare spectral estimate, and pseudo spectrum, using parametric and

metric techniques. Som e of the spectral analysis methods in cluded

olbox are Burg, covariance, eigenvector, Thomson multitaper,

ogram, Welch, and Yule-Walker. Other functions are available for

ing the average power of a power spectral density, computing a

ded spectrum, and shifting the DC component to the center of a

rum.

nalysis, and Implementation

r filter design functions included in the toolbox are FIR and

Transformations

unctions

Trans

The t the F

Sta

The cov

forms

oolbox includes a variety of transforms a nd inverse transforms, including

ourier, chirp-Z, discrete cosine, Goertzel, Hilbert, and Walsh-Hadamard.

tistical Signal Processing

toolbox has functions for computing correlation, cross-correlation,

ariance, and autocorrelation.

1-5

1 Overview

Parametric Mode

The toolbox incl Burg, covarianc The toolbox als or discrete-ti

Linear Predic

The toolbox h converting b coefficient

Multirate S

The toolbo including interpola

tion.

Waveform

The tool wavefor monopul voltage Sinuso

box has functions to generate many types of periodic and aperiodic

ms, including chirp, Dirichlet function, Gaussian RF pulse, Gaussian

se, pulse train, rectangle, sawtooth, sinc, square wave, triangle, and

-controlled oscillator. See “Waveform G eneration: Time Vectors and

ids” on page 2-4 for more information.

udes these methods for autoregressive parametric modeling: e, Yule-Walker, and Steiglitz-McBride (for ARMA modeling).

o has functions for fitting a frequency response to an analog

me filter.

tion

as functions for computing linear prediction coefficients and for

etween autorcorrelations and prediction polynomials, reflection

s, and line spectral frequencies.

ignal Processing

x includes a number of functions for multirate signal processing,

decimation, up- and downsampling, resampling, and spline

Generation

ling

1-6

Other O

A numb are ce

us plotting methods.

vario

perations

er of other operations are also available in the toolbox. Some of these

pstral analysis, modulation, demodulation, Slepian sequences, and

Interactive Tools

The power of Signal Processing Toolbox software is greatly enhanced by its easy-to-use interactive tools.

Interactive Tools

• The Filter Design and Analysis Tool (

a comprehensive collection of features for addressing filter design. Both FDAtool and filterbuilder offer seamless access to the additional filter design methods, quantization features, C-code generation and other enhanced filtering features of the Filter Design Toolbox product when that product is installed. If you have the Filter D esign HDL Coder™ product installed, you can also generate HDL code from both FDATool and filterbuidler.

• The Filter Visualization Tool (

for viewing, annotating, and printing filter response plots.

• The Signal Processing Tool (

for signal viewing, filter design, and spectral analysis.

• The Window Design and Analysis Tool (

for designing and comparing spectral windows.

• The Window Visualization Tool (

for viewing, annotating, and printing window plots.

fvtool) provides a graphical environment

sptool) provides a rich graphical environment

fdatool)andfil terb uilder provide

wintool)providesanenvironment

wvtool) provides a graphical environment

1-7

1 Overview

Extensibility

In this section...

“Expanding the Toolbox” on page 1-8

“Expanding the Toolbox with Other Toolboxes” on page 1-8

Expanding the Toolbox

One of the most important features of the MATLAB environment is that it is extensible. MATLAB lets you create your own programs for research, design, or engineering of signal processing systems. Simply copy the files provided with the Signal Processing Toolbox product and modify them as needed, or create new functions to expand the functionality of the toolbox.

Expanding the Toolbox with Other Toolboxes

A number of other MATLAB products expand and enhance the Signal Processing Toolbox product. These include:

1-8

• Filter Design Too lbox — Toolbox for advanced filter design

• Signal Processing Blockset™ — Product based on Simulink

and analyzing signal processing models

• Fixed-Point Toolbox — Toolbox for using fixed-point arithmetic

• Filter Design HDL Coder — Toolbox for creating and exporting HDL code

• Embedded IDE Link™ — Toolbox for working with embedded software on

Texas Instruments™ DSPs

for creating

Finding More Information

This Getting Started guide provides an introduction to Signal Processing Toolbox software and examples, which give you a quick start at using some of the commands a nd graphical user interfaces. It is assumed that you have basic knowledge and understanding of signals and systems, including such topics as filter and linear system theory and basic Fourier analysis.

More detailed and a dvanced information on using the toolbox is available in theonlinehelpsystembytyping viewing the documentation on the MathWorks Web site ( The toolbox also includes a number of introductory and advanced demos which you can access by typing

Finding More Information

doc at the MATLAB command line or by

www.mathworks.com).

demos at the MATLA B command line.

1-9

1 Overview

1-10

Basic Signal Processing Concepts

• “Representing Signals” on page 2-2

• “Waveform Generation: Time Vectors and Sinusoids” on page 2-4

• “Working with Data” on page 2-12

• “Selected Bibliography” on page 2-14

2 Basic Signal Processing Concepts

Representing Signals

In this section...

“Numeric Arrays” on page 2-2

“Vector Representation” on page 2-2

Numeric Arrays

The central data construct in the MATLAB environment is the numeric array, an ordered collection of real or complex numeric data with two or more

dimensions. The basic data objects of signal processing (one-dimensional signals or sequences, multichannel signals, and two-dimensional signals) are all naturally suited to array representation.

Vector Representation

MATLAB represents ordinary one-dim ensio n al sampled data signals, or sequences, as vectors.Vectorsare1-by-n or n-by-1 arrays, where n is the number of samples in the sequence. One way to introduce a sequence is to enter it as a list of elements at the command prompt. The statement

2-2

x = [4 3 7 -9 1];

creates a simple five-element real sequence in a row vector. Transposition turns the sequence into a column vector

x=x';

resulting in

4 3 7

-9 1

Column orientation is preferable for single channel signals becau se it e xten ds naturally to the multichannel case. For multichannel data, each column of a

Representing Signals

matrix represents one channel. Each row of such a matrix then corresponds to a sample point. A three-channel signal that consists of

y = [x 2*x x/pi]

x, 2x,andx/π is

This results in

4.0000 8.0000 1.2732

3.0000 6.0000 0.9549

7.0000 14.0000 2.2282

-9.0000 -18.0000 -2 .864 8

1.0000 2.0000 0.3183

If the sequence has complex-valued elements, the transpose operator takes theconjugateofthesequenceelements. Totransformacomplex-valued row vector into a column vector without taking conjugates, use the

.' or

non-conjugate transpose:

x=[1-i 3+i 2+3*i 4-2*i]; %1X4 x=x.'; %4X1

2-3

2 Basic Signal Processing Concepts

Waveform Generation: Time Vectors and Sinusoids

In this section...

“Time Vectors” o n page 2-4

“Common Sequences: Unit Impulse, Unit Step, and Unit Ramp” on page 2-5

“Multichannel Signals” on page 2-5

“Common Periodic Waveforms” on page 2 -6

“Common Aperiodic Waveforms” on page 2-7

“The pulstran Function” on page 2-8

“The Sinc Function” on page 2 -9

“The Dirichlet Function” on page 2-10

Time Vectors

Most toolbox functions require you to begin with a vector representing a time base. Consider generating data with a 1000 Hz sample frequency, for example. An appropriate time vector is

2-4

t = (0:0.001:1)';

where the MATLAB colon operator creates a 1001-ele m ent row vector that represents time running from 0 to 1 s in steps of 1 ms. The transpose operator

(') changes the row vector into a column; the semicolon (;)tellsMATLABto

compute, but not display the result.

Given 50 Hz and one at 120 Hz with twice the amplitude.

The new variable y,formedfromvectort,isalso1001elementslong. You can add normally distributed white noise to the signal and plot the first 50 points using

t, you can create a sample signal y consisting of two sinusoids, one at

y = sin(2*pi*50*t) + 2*si n(2* pi*120*t);

randn('state',0); yn = y + 0.5*randn(size(t)); plot(t(1:50),yn(1:50))

Waveform Generation: Time Vectors and Sinusoids

Common Sequences: Unit I mpulse, Unit Step, and Unit Ramp

Since MATLAB is a programming language, an endless variety of different signals is possible. Here are some statements that generate several commonly used sequences, including the unit impulse, unit step, and unit ramp functions:

t = (0:0.001:1)'; imp= [1; zeros(99,1)]; % Impulse unit_step = ones(100,1); % Step (with 0 initial cond.) ramp_sig= t; % Ramp quad_sig=t.^2; % Quadratic sq_wave = square(4*pi*t); % Square wave with period 0.5

All of these sequences are column vectors.Thelastthreeinherittheirshapes from

Multichannel Signals

Use standard MATLAB array syntax to work with multichannel signals. For example, a multichannel signal consisting of the last three signals generated above is

2-5

2 Basic Signal Processing Concepts

z = [ramp_sig quad_sig sq_wave];

You can generate a multichannel unit sample function using the outer product operator. For example, a six-element column vector whose first element is one, and whose remaining five elements are zeros, is

a = [1 zeros(1,5)]';

To duplicate column vector a into a matrix without performing any multiplication, use the MATLAB

c = a(:,ones(1,3));

Common Periodic Waveforms

The toolbox provides functions for generating widely used periodic waveforms:

•

sawtooth generates a sawtooth wave with peaks at ±1 and a period of

2π.Anoptional which the signal’s maximum occurs.

colon operator and the ones function:

width parameter specifies a fractional multiple of 2π at

2-6

•

square generates a square wave with a period of 2π. An optional parameter

specifies duty cycle, the percent of the period for which the sig n al is positive.

Togenerate1.5sofa50Hzsawtoothwavewithasamplerateof10kHzand plot 0.2 s of the generated waveform, use

fs = 10000; t = 0:1/fs:1.5; x = sawtooth(2*pi*50*t); plot(t,x), axis([0 0.2 -1 1])

Waveform Generation: Time Vectors and Sinusoids

Common Aperiodic Waveforms

The toolbox also provides functions for generating several widely used aperiodic waveforms:

•

gauspuls generates a Gaussian-modulated sinusoidal pulse with a

specified time, center frequency, and fractional bandwidth. Optional parameters return in-phase and quadrature pulses, the RF signal envelope, and the cutoff time for the trailing pulse envelope.

•

chirp generates a linear, log, or quadratic swept-frequency cosine signal.

An optional parameter specifies alternative sweep methods. An optional parameter

phi allows initial phase to be specified in degrees.

To compute 2 s of a linear chirp signal with a sample rate of 1 kHz, that starts at DC and crosses 150 Hz at 1 s, use

t = 0:1/1000:2; y = chirp(t,0,1,150);

To plot the spectrogram, use

spectrogram(y,256,250,256,1000,'yaxis')

2-7

2 Basic Signal Processing Concepts

The pulstr

The pulst sampled p consisti The pulse length o to each p repetit the specif

ng of the sum of multiple delayed interpolations of a Gaussian pulse.

ion in column 2. The pulse train is constructed by passing the name of

spuls

gau

y a 10 kHz Gaussian pulse with 50% bandwidth:

T = 0:1/50E3:10E-3; D = [0:1/1E3:10E-3;0.8.^(0:10)]'; Y = pulstran(T,D,'gauspuls',10E3,0.5); plot(T,Y)

an Function

ran

function generates pulse trains from either continuous or

rototype pulses. The following example generates a pulse train

train is defined to have a sample rate of 50 kHz, a pulse train

f 10 ms, and a pulse repetition rate of 1 kHz;

ulse repetition in column 1 and an optional attenuation for each

function to pulstran, along with additional parameters that

D specifies the delay

2-8

Waveform Generation: Time Vectors and Sinusoids

The Sinc Function

The sinc function computes the mathematical sinc function for an input vector or matrix the inverse Fourier transform of the rectangular pulse in frequency centered at zero of width 2π and he ight 1. The following equation defines the sinc function:

x. Viewed as a function of time, or space, the sinc function is

sin( )





The sinc function has a value of 1 whenx is equal to zero, and a value of

sin( )



forallotherelementsofx.

To plot the sinc function for a linearly spaced vector with values ranging from

-5 to 5, use the following commands:

x = linspace(-5,5);



∫



−



2-9

2 Basic Signal Processing Concepts

y = sinc(x); plot(x,y)

2-10

The Dirichlet Function

The toolbox function diric computes the Dirichlet function, sometimes called the periodic sinc or aliased sinc function, for an input vector or matrix Dirichlet function is

sin( / )

(

−

1))

xkk

, ,,,,...

≠=±±±

20123

, ,,,,...xkk==±±±

20123



diric( )

⎧ ⎪

⎨ ⎪

()

−

⎩

where N is a user-specified positive integer. For N odd, the Dirichlet function has a period of 2π;forN even, its period is 4π. The magnitude of this function is (1/N) times the magnitude of the discrete-time Fourier transform of the N-point rectangular window.

To plot the Dirichle t function over the range 0 to 4π for

x = linspace(0,4*pi,300);

N = 7 an d N = 8,use

x.The

plot(x,diric(x,7)); axis tight; plot(x,diric(x,8)); axis tight;

Waveform Generation: Time Vectors and Sinusoids

2-11

2 Basic Signal Processing Concepts

Working with Data

In this section...

“Importing Data from Within the MATLAB Environment” on page 2-12

“Importing Data from Outside the MATLAB Environment” on page 2-12

“Converting Data in to a MAT-File” on page 2-12

“Exporting Data” on page 2-13

“Data Precision” on page 2-13

Importing Data from Within the MATLAB Environment

Theexamplesintheprecedingsectionsobtaindatainoneoftwoways:

• By direct input, that is, entering the data manually at the keyboard

• By using a MATLAB or toolbox function, such as

square,orsinc

sin, cos, sawtooth,

Importing Data from Outside the MATLAB Environment

Some applications, however, may need to import data from outside MATLAB. Depending on your data format, you can do this in the following ways:

• Load data from an ASCII file or MAT-file with the MATLAB

command.

• Read the data into MATLAB with a low-level file I/O function, such as

fopen, fread,andfscanf.

• Develop a MEX-file to read the data.

load

Converting Data into a MAT-File

Other resources are also useful, such as a high-level language program (in Fortran or C, for example) that conv erts your data into MAT-file format. See the “Manually Converting Data Passed to Functions” documentation for details. MATLAB reads such files using the

load command.

2-12

Working with Data

Exporting Data

Similar techniques are available for exporting data generated within MATLAB. See the “Importing and Exporting Data” documentation for more details.

Data Precision

All Signal Processing Toolbox functions accept double-precision inputs. If you input single-precision floating-point or integer data types, you should not expect to receive correct results and in many cases, an error will occur. Filter Design Toolbox and Fixed-Point Toolbox products enable single-precision floating-point and fixed-point support for most

dfilt structures.

2-13

2 Basic Signal Processing Concepts

Selected Bibliography

Algorithm development for Signal Processing Toolbox functions has drawn heavily upon the references listed below. All are recommended to the interested reader who needs to know more about signal processing than is covered in this manual.

[1] Crochiere, R.E., and L.R. Rabiner. Multi-Rate Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1 98 3 . pp. 88-91.

[2] IEEE. Programs for Digital Signal Processing. IEEE Press. New York: John Wiley & Sons, 1979.

[3] Jackson, L.B. Digital Filters and Signal Processing.ThirdEd.Boston: Kluwer Academic Publishers, 1989.

[4] Kay, S.M. Modern Spectral Estimation. Englewood Cliffs, NJ: Prentice Hall, 1988.

2-14

[5] Op penheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1989.

[6] Parks, T.W., and C.S. Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987.

[7] P erci va l, D.B., and A .T. Walden. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge: Cambridge University Press, 1993.

[8] Pratt,W.K. D igital Image Processing. New York: John Wiley & Sons, 1991.

[9] Proakis, J.G., and D.G. Manolakis. Digital Signal Processing: Principles, Algorithms, and Applications. Upper Saddle River, NJ: Prentice Hall, 1996.

[10] Rabiner, L.R., and B. Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1975.

[11] Welch, P.D. “The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified

Selected Bibliography

Periodograms.” IEEE Trans. A udio Electroacoust.Vol.AU-15(June1967). Pgs. 70-73.

2-15

2 Basic Signal Processing Concepts

2-16

Filter Design with Fdesign and Filterbuilder

• “Filter Design Process Overview” on page 3-2

• “Basic Filter Design Process” on page 3-4

• “Using Filterbuilder to Design a Filter” on page 3-9

3 Filter Design with Fdesign and Filterbuilder

Filter Design Process Overview

Note You must have the Signal Processing Toolbox installed to use fdesign

and fil terb uilder. A dvanced capabilities are available if your installation additionally includes the Filter Design Toolbox license. You can verify the presence of both toolboxes by typing

Filter design through user-defined specifications is the core of the fdesign approach. This specification-centric approach places less emphasis on the choice of specific filter algorithms, and more emphasis on performance during the design a good working filter. For example, you can take a given set of design parameters for the filter, such as a stopband frequency, a passband frequency, and a stopband attenuation, and— using these parameters— design a specification object for the filter. You can then implement the filter using this specification object. Using this approach, it is also possible to compare different algorithms as applied to a set of specifications.

ver at the command prompt.

3-2

There are two distinct objects involved in filter design:

• Specification Object — Captures the required design parameters of a filter

• Implementation Object — Describes the designed filter; includes the array

of coefficients and the filter structure

The distinction between these two objects is at the core of the filter design methodology. The basic attributes of each of these objects are outlined in the following table.

fication Objec t

Speci

-level specification

High

rithmic properties

Algo

You can run the code in the following examples from the Help browser (select the code, right-click the selection, and choose Evaluate Selection from the context menu), or you can enter the code on the MATLAB command line. Before you begin this example, start MATLAB and verify that you have installed the Signal Processing Toolbox software. If you wish to access the

Implementation Object

er coefficients

Filt

Filter structure

Filter Design Process Overview

full functionality of fdesign and filterbuilder, you should additionally obtain the Filter Des ign Toolbox software. You can verify the presence of these products by typing

ver at the command prompt.

3-3

3 Filter Design with Fdesign and Filterbuilder

Basic Filter Design Process

Use the following two steps to design a simple filter.

1 Create a filter specification object.

2 Design your filter.

Example — Design a Filter in Two Steps

Assume that you want to design a bandpass filter. Typically a bandpass filter is defined as shown in the following figure.

3-4

In this example, a sampling frequency of Fs = 48 kHz is used. This bandpass filter has the following specifications, specified here using MATLAB code:

A_stop1 = 60; % Attenuation in the first stopband = 60 dB F_stop1 = 8400; % Edge of the stopband = 8400 Hz F_pass1 = 10800; % Edge of the passband = 10800 Hz F_pass2 = 15600; % Closing edge of the passband = 15600 Hz F_stop2 = 18000; % Edge of the second stopband = 18000 Hz A_stop2 = 60; % Attenuation in the second stopband = 6 0 dB A_pass = 1; % Amount of ripple allowed in t he passband = 1 dB

In the followin g two steps, these spe c if ica t ion s are passed to the

fdesign.bandpass method as parameters.

Basic Filter Design Process

Step 1

To create a filter specification object, evaluate the following code at the MATLAB prompt:

d = fdesign.bandpass

Now, pass the filter specifications that correspond to the default

Specification — fst1,fp1,fp2,fst2,ast1 ,ap,ast2.Thisexampleadds fs as the final input argument to specify the sampling frequency of

48 kHz.

>> BandPassSpecObj = ...

fdesign.bandpass('Fst1,Fp1,Fp2,Fst2,Ast1,Ap,Ast2', ... F_stop1, F_pass1, F_pass2, F_stop2, A_stop1, A _pas s, ... A_stop2, 48000)

Note The order of the filter is not specified, allowing a degree of freedom for the algorithm design in order to achieve the specification. The design will be a minimum order design.

The specification parameters, such as Fstop1,areallgivendefault values when none are provided. You can change the values of the specification parameters after the filter specification object has been created. For example, if there are two values that need to be changed,

Fpass2 and Fstop2,usetheset command, which takes the object first,

and then the parameter value pairs. Evaluate the following code at the MATLAB prompt:

>> set(BandPassSpecObj, 'Fpass2', 15800, 'Fstop2', 18400)

BandPassSpecObj

is the new filter specification object which contains

all the required design parameters, including the filter type.

You may also change parameter values in filter specification objects by accessing them as if they were elements in a

>> BandPassSpecObj.Fpass2=15800;

struct array.

3-5

3 Filter Design with Fdesign and Filterbuilder

Step 2

Design the filter by using the design methods available for you specification object by calling the

designmethods function. For example, in this case, you can execute

the command

>> designmethods(BandPassSpecObj)

Design Methods for class fdesign.bandpass (Fst1,Fp1,Fp2,Fst2,Ast1,Ap,Ast2):

butter cheby1 cheby2 ellip equiripple kaiserwin

design command. You can access the

3-6

After choosing a design method use, you can evaluate the following at the MATLAB prompt (this example assumes you’ve chosen ’

>> BandPassFilt = design( BandPassSpecObj, 'equiripple')

BandPassFilt =

FilterStructure: 'Direct-Form FIR'

Arithmetic: 'double'

Numerator: [1x44 double]

PersistentMemory: false

equiripple’):

Basic Filter Design Process

Note If you do not specify a design method, a default method will be used. For example, you can execute the command

>> BandPassFilt = design( BandPassSpecObj)

BandPassFilt =

FilterStructure: 'Direct-Form FIR'

Arithmetic: 'double'

Numerator: [1x44 double]

PersistentMemory: false

and a design method will be selected automatically.

To check your work, you can plot the filter magnitude response using the Filter Visualization tool. Verify that all the design parameters are met:

>> fvtool(BandPassFilt) %plot the fil ter magnitude response

If you have the Filter Design Toolbox installed, the Filter Visualization tool produces the following figure with the dashed red lines indicating the transition bands and unity gain (0 in dB) over the passband. If you do not have the Filter Design Toolbox,thefigureappearswithout the dashed red lines.

3-7

3 Filter Design with Fdesign and Filterbuilder

With only the Signal Processing Toolbox software installed:

3-8

Using Filterbuilder to Design a Filter

Filterbuilder presents the option of designing a filter using a GUI dialog box as opposed to the command line instructions. You can use Filterbuilder to design the same bandpass filter designed in the previous section, “Basic Filter Design Process” on page 3-4

Example — Using Filterbuilder to Design a Simple Filter

To design the filter using FilterBuilder:

1 Type the following at the MATLA B prompt:

filterbuilder

The dialog box using the Filter Design Toolbox software appears a s follows:

Using Filterbuilder to Design a Filter

The dialog box using only the Signal Processing Toolbox software appears as follows:

3-9

3 Filter Design with Fdesign and Filterbuilder

2 Select Bandpass filter response from the list in the dialog box, and hit the

OK button. The following dialog box opens:

3-10

Using Filterbuilder to Design a Filter

3 Enter the correct frequencies for Fpass2 and Fstop2,asshowninthe

preceding figure, then click OK. Here the specification uses normalized frequency, so that the passband and stopband edges are expressed as a fraction of the Nyquist frequency (in this case, 48/2 kHz). The following message appears at the MATLAB prompt:

The variable 'Hbp' has be en exported to the command window.

IfyoudisplaytheWorkspacetab,asshowninthefollowingfigure,yousee the object

Hbp has been placed on your workspace.

3-11

3 Filter Design with Fdesign and Filterbuilder

4 To check your work, plot the filter magnitude response using the Filter

Visualization tool. Verify that all the design parameters are met:

fvtool(Hbp) %plot the fil ter magnitude response

3-12

Note that the dashed red line s on the preced ing figure will only appear if you are using the Filter Design Toolbox software.

Filter Design with the FDATool GUI

• “Introduction” on page 4-2

• “Designing the Filter” on page 4-3

• “Analyzing the Filter” on page 4-8

• “Designing Additional Filters” on page 4-10

• “Viewing and Annotating the Filter” on page 4-11

• “Exporting Filters from FDATool” on page 4-17

• “Designing Filters using Command Line Functions” on page 4-21

• “Where to Find More Information” on page 4-24

4 Filter Design with the FDATool GUI

Introduction

This section describes how to graphically design and implement digital filters using the Signal Processing Toolbox FDATool GUI. Filter design is the process of creating the filter coefficients to meet specific frequency specifications. Filter implementation involves choosing and applying a particular filter structure to those coefficients. Only after both design and implementation have been performed can your data be filtered.

This section includes a brief discussion o f applying the completed filter design and filter implementation using MATLAB command line functions, such as

filter.

For an interactive FDATool demo, type line, and select Toolboxes. Expand the tree, scroll down, and select Signal

Processing Toolbox. Under Filter Design and Analysis, click Introduction to Filter Design and Analysis Tool.

demos at the MATLAB command

4-2

Designing the Filter

This section is a step-by-step introduction to using the Filter Design and Analysis Tool (FDAToo l) to design an octave-band filter. An octave is the interval between two frequencies having a ratio of 2:1. An octave-band filter is a bandpass filter with high cutoff frequency approximately twice that of the low cutoff frequency. The class of an octave filter is determined by its allowable passband ripple and its stopband attenuation. Refer to the ANSI S1.11–2004 standard for more information. For more information on designing filters, see “FDATool: A Filter Design and Analysis GUI” in the Signal Processin g Toolbox User’s Guide. (N o te that you ca n also access FDATool from SPTool).

1 Start FDATool from the M AT LAB command line.

fdatool

The FDATool dialog opens with a default filter. Its filter information is summarized in the upper left (Current Filter Information) and its filter specifications are depicted in the upper right. In addition to displaying filter specification, this upper right pane di splays fi lter responses and filter coefficients.

Designing the Filter

4-3

4 Filter Design with the FDATool GUI

The bottom half of FDATool shows the Filter Design panel, where you specify the filter parameters. Other panels, such as Import filter from workspace and Pole/Zero Editor, which you access with the buttons on the lower left, are also displayed in this area. If you have other products installed, you may see additional buttons.

Note that w h en you open FDATool, Design Filter is not enabled. You must make a change to the default filter design in order to enable Design Filter.Thisistrueeachtimeyouwanttochangethefilterdesign. Changes to radio button items or drop down menu items such as those under Response Type or Filter Order enable Design Filter immediately. Changes to specifications in text boxes such as Fs, Fpass,andFstop require you to click outside the text b ox to enable Design Filter.

2 In the Response Type pane, select Bandpass.

3 In the Design Method pane, select IIR,andthenselectButte rwo rth

from the selection list.

4-4

4 For the Filter Order, select Specify order, and then enter 6.

5 Set the Frequency Specifications as follows:

Parameter Setting Description

Units

Fc1

48000

Units for the parameters

Sampling frequency

First cutoff frequency (i.e., the frequency preceding the passband at which the magnitude response is 3 dB below the passband gain)

Fc2

Second cutoff frequency (i.e., the frequency following the passband at which the magnitude response is 3 dB below the passband gain)

Designing the Filter

6 After specifying the filter design parameters, click the Design Filter

button at the bottom of the design panel to compute the filter coefficients. The display u pdates t o show the magnitude response of the designed filter.

4-5

4 Filter Design with the FDATool GUI

Notice that the Design Filter button is disabled after you compute the coefficients for your filter design. This button is enabled again if you make any changes to the filter specifications.

7 Click the Store Filter button.

4-6

8 In the Store Filter dialog, change the filter name to Bandpass

Butterworth-1

and click OK to save the filter in the Filter Manager.

Designing the Filter

‘

4-7

4 Filter Design with the FDATool GUI

Analyzing the Filter

After designing the filter, you can view the following filter responses in the display region by clicking on the associated toolbar button or by selecting the desired response from the Analysis menu.

Response Toolbar Button Image

Filter specifications

Magnitude response

Phase response

Magnitude

Group delay

Phase delay

Impulse response

Step response

Pole-zero plot

Filter

Filter information

Note Other analyses are available if you have the Filter Design Toolbox product installed.

1 Examine the displayed magnitude response of the filter.

and Phase responses

coefficients

4-8

Analyzing the Filter

2 Display other responses, as desired. Click the appropriate buttons, show n

in the table above or select the desired response from the Analysis menu.

3 Click the Filter coefficients button to display the filter coefficients.

4-9

4 Filter Design with the FDATool GUI

Designing Additional Filters

You have designed one of the bands of an octave filter bank. This section shows you how to design and save the other nine bands. The following table defines the parameters for the remaining bands. Note that all of the bands use these parameters: Bandpass, IIR – Butterworth , order =

48000 Hz .

6, Fs =

Fc1

45 89 Bandpass Butterworth-2

89 178 Bandpass Butterworth-3

178 355 Bandpass Butterworth-4

355 708 Bandpass Butterworth-5

708 1413 Bandpass Butterworth-6

1413 2818 Bandpass Butterworth-7

2818 5623 Bandpass Butterworth-8

5623 11220 Bandpass Butterworth-9

11220 22387 Bandpass Butterworth-10

1 Using the parameters listed in the table above, for each table row, set the

Fc2

Filter Nam e

appropriate the Fc1 and Fc2 values.

2 Design the filter by clicking the Design Filter button.

lick Store Filter to save the filter.

3 C

4 Changethenametotheappropriatefilternameshowninthetableabove.

5 Repeat these steps until all 10 filters are designed and stored.

4-10

Viewing and Annotating the Filter

In this section...

“Viewing the Filter in FVTool” on page 4-11

“Using FVTool for Annotation” on page 4-15

Viewing the Filter in FVTool

This section teaches you how to use the Filter Visualization Tool (FVTool) to view the octave-band filter. It also describes how to annotate your filter.

1 Click the Filter Manager button to display the Filter Manager, which

lists your saved filters.

Viewing and Annotating the Filter

2 Press Ctrl+click on each filter name to select all the filters, and then

click FVTool. FVTool opens with the filter responses overlaid f or easy comparison. (If you want to view a single filter in FVTool, click the Full

4-11

4 Filter Design with the FDATool GUI

View Analysis button when that filter is shown in the FDATool display panel or select View > Filter Visualization Tool).

3 Change the x-axis scale to logarithmic by selecting Analysis > Analysis

Parameters to display the A n alysis Parameters dialog.

4 Change the Frequency Scale to Log.

4-12

5 Click OK.

Viewing and Annotating the Filter

6 Click the Legend button to turn on the legend, which you can drag to

the desired location.

4-13

4 Filter Design with the FDATool GUI

4-14

7 Click the Legend button again to turn off the legend.

Use the Zoom button passbands to zoom in.

and drag a rectangle a round the first few

Viewing and Annotating the Filter

8 Click the Restore Default View button to return to the full view.

9 Display other responses, as desired. (The FVTool Analysis toolbar buttons

and Analysis menu are the same as in FDATool. See “Analyzing the Filter” on page 4-8 for descriptions of the buttons.

Using FVTool for Annotation

FVTool is also useful for doing further analysis, adding annotations, and printing. Available annotations include adding rectangles, text boxes, arrows and lines, and adding data tips.

For a demo about FVTool, type and select Toolboxes. Expand the tree, scroll down, and select Signal

Processing Toolbox. Under Filter Design and Analysis, click Filter Analysis using FVTool and its API.

demos at the MATLAB command line,

4-15

4 Filter Design with the FDATool GUI

Note Do not close F DATool a t this ti m e. You will use it in future sections.

1 Use the toolbar buttons to annotate your response plot. Add a line by

clicking one of the line buttons, and then use your mouse to draw the line on your plot.

2 Add a data tip by clicking on a plot at the desired point. The data tip shows

the frequency and magnitude at that point.

4-16

3 Close FVTool by selecting File > Close.

Exporting Filters from FDATool

FDATool provides a simple way to create filter objects (dfilts) from your filter designs. This is particularly useful for saving your filter design to the MATLAB workspace for use with command line functions. You can also save your filters to M-files using File > Generate M-file to run in scripts or batch files.

1 In FDATool, click Filter Manager and highlight only the Bandpass

Butterworth-1

2 Select File > Export.

3 Set Export to to Workspace.SetExport as to Objects.InDiscrete

Filter type an

Hd1 dfilt object in the w orkspace.

filter.

Hd1.ClickExport to export the first filter in your filter bank to

Exporting Filters from FDATool

eat steps 1 through 3 for each of the remaining nine filters. Highlight

4 Rep

h filter individually to make it the active filter and change the Discrete

eac

lter name to match the filter number. When you finish you will have

dfilt objects in the workspace.

4-17

4 Filter Design with the FDATool GUI

5 Close FDATool by selecting File > Close.

6 On the MATLAB command line, verify that your objects were exported by

using the

whos

Filtering with dfilt

1 Type the following on the MATLAB co mmand line to concatenate your

filter bank filter objects into a single

whos command.

Name Size Bytes Class Attributes

Hd1 1x1 dfilt.df2sos Hd10 1x1 dfilt.df2sos Hd2 1x1 dfilt.df2sos Hd3 1x1 dfilt.df2sos Hd4 1x1 dfilt.df2sos Hd5 1x1 dfilt.df2sos Hd6 1x1 dfilt.df2sos Hd7 1x1 dfilt.df2sos Hd8 1x1 dfilt.df2sos Hd9 1x1 dfilt.df2sos

dfilt object.

4-18

Hd = [Hd1 Hd2 Hd3 Hd4 Hd5 Hd6 Hd7 Hd8 Hd9 Hd10];

2 To view the first filter, type Hd(1).

Hd(1)

ans =

FilterStructure: 'Direct-Form II, Second-Order Sections'

sosMatrix: [3x6 double]

ScaleValues: [3.40097054256801e-009;1;1;1]

PersistentMemory: false

3 A number of methods can be used to view and manipulate the Hd1 dfilt

object. Try the

info(Hd1) % Displays filter information

Discrete-Time IIR Filter ( real)

info command:

Exporting Filters from FDATool

------------------------------Filter Structure : Direct-Form II, Second-Order Sections Number of Sections : 3 Stable : Yes Linear Phase : No

4 You can open FVTool from the MATLAB command line and specify display

parameters as follows.

F = fvtool(Hd,'Analysis','magnitude') % Open FVTool with

% magnitude display

set(F,'FrequencyScale','Log') % Chang e to log scale

This pro duces the same display as step 5 of “Viewing the Filter in FVTool” on page 4-11 above.

5 Now using the MATLAB command line, create some discrete white

Gaussian noise data, which you can then filter using the filter bank.

rand; % Initialize random numbe r generator Nx = 100000; % Number of noise data points xw = randn(Nx,1); % Create white noise for i=1:10,

yw(:,i)=filter(Hd(i),xw); % Filter the white n oise through

end % the entire filter bank.

% (:,i) means all rows of column i

6 Plot the filtered data.

plot(yw)

4-19

4 Filter Design with the FDATool GUI

4-20

Designing Filters using Command Line Functions

Beginning with R2009a, users can specify and design filters at the command line using powerful and efficient way to specify and implement digital filters. With

fdesign and design, digital filter design is a two-step process. In the first

step, the user specifies the desired characteristics of the filter and saves those specifications in a Filter Specification Object. In the second step, the user implements the filter as a dfilt object, which can then be used to filter data. As an example of this two-step process, we will design a lowpass filter for data sampled at 20 kHz. The desired passband frequency is 1 kHz with a stopband frequency of 1.2 kHz. We will limit the passband ripple to 1 dB and require 60 dB of attenuation between the passband and stopband frequencies.

1 To specify the filter use fdesign.lowpasswith the parameters given above.

You can copy and paste the following code at the MATLAB command prompt.

fdesign and design. The use of fdesign and design provides a

d=fdesign.lowpass('Fp,Fst,Ap,Ast',1000,1200,1,60,20000);

2 To design the filter use design with the appropriate design methods. In

this example, we create FIR equiripple and IIR Butterworth designs and compare the filters’ magnitude responses.

Hd1=design(d,'equiripple'); %FIR equiripple design Hd2=design(d,'butter'); %Butterworth design Hd=[Hd1 Hd2];%filter object with both designs %compare filters fvtool(Hd,'legend','on'); axis([0 2 -70 10])

4-21

4 Filter Design with the FDATool GUI

4-22

To determine which filter design methods are available for a given Filter Specification Object, use designmethods. For the lowpass filter example above, you have a choice of two FIR and four IIR digital filter designs:

nmethods(d)

desig

n Methods for class fdesign.lowpass (Fp,Fst,Ap,A st) :

Desig

butt

cheb

ell

iripple

equ

serwin

kai

Designing Filters using Command Line Functions

For more d etailed information on filter specification and implementation objects see “Designing a Filter in Fdesign — Process Overview”.

4-23

4 Filter Design with the FDATool GUI

Where to Find More Information

The previous sections des cribed how to use the fundamental features of FDATool, FVTool, and the advanced information and more complex examples, refer to other sections of Signal Processing Toolbox online help or Signal Processing Toolbox documentation available on The MathWorks Web site (

fdesign and design commands. For more

www.mathworks.com).

4-24

Spectral Analysis

• “Introduction” on page 5-2

• “Creating a Spectral Analysis Object” on page 5-4

• “Producing a PSD Estimate” on page 5-6

• “Changing Spectral Analysis Object P roperty Values” on page 5-8

• “Measuring Signal Power” on page 5-13

5 Spectral Analysis

Introduction

In this section...

“Spectral Estimators” on page 5-2

“Spectral Analysis Algorithms” on page 5-2

“Spectral Analysis Objects” on page 5-3

Spectral Estimators

Spectral analysis includes three types of spectral estimators — power spectral density (PSD), mean-square spectrum (MSS) and pseudo spectrum.

• Power spectral density (

has power/frequency units.

• Mean-square (power) spectrum (

frequency.

• Pseudospectrum (

not have any units.

psd) measures power per unit of frequency and

msspectrum) measures power at a specific

pseudospectrum) returns a pseudo spectrum that does

Spectral Analysis Algorithms

Signal Processing Toolbox software provides several algorithms to compute of spectral estimates. The following table indicates which algorithms are available to compute each type of estimator and produce a spectrum object. For general information on objects, see Object-Oriented Programming.For details on spectrum objects, see the

spectrum reference page.

5-2

Introduction

Spectral

Algorithms

Estimator

Power spectral density (

psd)

Mean-square spectrum (

msspectrum)

Pseudo spectrum (

pseudospectrum)

Burg (

spectrum.burg),

Covariance ( Modified covariance ( Thomson multitaper method (MTM) ( Periodogram ( Welch ( Yule-Walker autoregressive ( See also

Periodogram ( Welch ( See also

Eigenvector (

spectrum.cov),

spectrum.mcov),

spectrum.mtm),

spectrum.periodogram),

spectrum.welch),

spectrum.yulear)

dspdata.psd

spectrum.periodogram),

spectrum.welch)

dspdata.msspectrum

spectrum.eigenvector),

MUSIC (Multiple Signal Classification) (

spectrum.music)

Mathworks SIGNAL PROCESSING TOOLBOX 6 Installation Guide

Specifications and Main Features

Frequently Asked Questions

User Manual