     [file version.texi not included]

This is a manual for GNU C-Graph version VERSION, a tool for learning about convolution.

Copyright © 1982, 1983, 1996, 2008, 2009, 2010, 2011 Adrienne Gaye Thompson, Sole Author. GNU C-Graph version 2.0. Derived from BSc. dissertation "Interactive Computer Package Demonstrating: Sampling Convolution and the FFT", University of Aberdeen, Scotland (1983). For the code from the dissertation, visit <http://codeartnow.com/law-project>.

     Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts.  A copy of the license is included in the section entitled ``GNU Free Documentation License''.

for  ALL the VICTIMS of APARTHEID STRUGGLING to be FREE

and

to REGNIER

You're sending me discrete signals from across the room,

I respond on impulse, reflecting on the sampling of events

That were a dichotomy from the day you left your mother's womb;

Multiplied in frequency, integrated in time, a weighted confluence

Of sliding, shifting trains of thought, alternative messages under transformation;

Counterpoint, duality, involution, contradistinction without confusion;

Independence in summation. Silence - this is convoluted conversation.

C-Graph:
********

This is a manual for GNU C-Graph version VERSION, a tool for learning about convolution.

Foreword
********

From the shadow cast by light to the echo in a cave convolution, like the ubiquitous Fibonacci series, is a mathematical description of naturally ocurring physical phenomena in any linear, time-invariant system capable of responding to an input signal. Today, convolution - the combination of two signals to produce a third - has wide ranging applications. Edge detection in computer vision, algorithms for robot motion, signal and image processing, crystallography, statistics and probability theory, differential equations, linear algebra, numerical analysis, and even recent innovations in music production - all utilise techniques involving convolution.

GNU C-Graph (for Convolution Graph) is a tool for visualizing the convolution of two signals. The package is a reproduction of the Fortran 77 program in my BSc. Electrical Engineering (Honours) dissertation "Interactive Computer Program Demonstrating: Sampling Convolution and the FFT", University of Aberdeen, Scotland, 1983. In this version I have included pulses, scaling of the signals, and error-handling - features that were not part of my original Thesis.

Whether student engineer or scientist, aspiring special-effects animator or roboticist, GNU C-Graph will help you find the adventure in the mathematics of convolution.

-- Adrienne Gaye Thompson

Overview
********

About
=====

GNU C-Graph computes the linear convolution of two signals in the time domain then compares their circular convolution by demonstrating the convolution theorem - convolution of two signals in the time domain corresponds to multiplication in the frequency domain. Each signal is modelled by a register of discrete values simulating samples of a signal, and the discrete Fourier transform (DFT) computed by means of the Fast Fourier Transform (FFT). See Appendix A, for an explanation of the convolution theorem.

GNU C-Graph is interactive, prompting the user to enter single character or numerical values from the keyboard, thereby dispensing with the learning curve for coding formulae. The user chooses 2 from a menu of 8 signal types, and up to 5 parameters to define the waveforms. The signals chosen may be periodic, aperiodic, or pulses. C-Graph then plots 3 graphs:

 1. The time domain representation of both signals;

 2. Their Fourier transforms;

 3. A comparison of their linear and circular convolution.

See A C-Graph Session, for a typical C-Graph session.

GNU C-Graph will be useful to students of signal theory in the study of convolution and spectral analysis. This version (2.0) uses a simple FFT written by Arthur Wouk and converted to Fortran 90 by Alan Miller (see Appendix B).

Required Software
=================

GNU C-Graph is written in contemporary Fortran. The package runs on GNU/Linux, was developed with GFortan and G95, and uses Gnuplot 4.2 as well as Image Magick 6.6.

Experienced users wishing to use other compilers may supply the necessary command-line options to configure during installation. See the file INSTALL for basic installation instructions.

Invoking C-Graph
****************

To run GNU C-Graph, open up a terminal in X and type c-graph. C-Graph supports the following options:

--dedicate
-d
     Print the dedication and exit.

--help
-h
     Print a summary of the command line options and exit.

--no-splash
-n
     Invoke GNU C-Graph with no splash screen.

--version
-v
     Print the version number and licensing information of GNU C-Graph, then exit.

The Signals
***********

The Menu
========

C-Graph presents the following menu of signals from which the user chooses 2:

SIGNAL   CODE
Sine
A
Cosine
B
Triangle
C
Square
D
Sawtooth
E
Exponential
F
Ramp
G
Step
H

Signals A to E are periodic, while F, G, and H are aperiodic. Pulses may also be chosen; these are a half period in duration 1/2f, where f is the frequency of the corresponding cyclical waveform.

Parameters
==========

The user enters up to 5 parameters to generate the signals, their FFTs and their convolution:

 1. The number of samples N

 2. The code for the signal A to H

 3. The wave/pulse parameter w or p

 4. The frequency f

 5. The scaling coefficient sc

Both signals are constructed from the same number of samples N. If the user chooses a periodic signal, then he/she is prompted to select either the cyclical waveform or a derived pulse, i.e., w or p. For each periodic signal chosen the user is prompted to enter its frequency f and a scale factor sc.

Pulses are monophasic and are defined on half the period of the modulus of the corresponding periodic waveform.

Defaults & Error Handling
=========================

If the required parameter is a number and the user has erroneously entered character data, C-Graph generates an error message and gives the user another try to enter a number. Otherwise, for input outside the expected ranges C-Graph assumes default values.

Number of samples N
     N must lie in the range [64, 1024]. Values entered outside of this range will default to 512. N is defined to be a power of 2. If the user enters a value that is not a power of 2 C-Graph will choose the nearest power of 2.

Signal code A to H
     For input outside the range A to H, the default codes are C for the first signal, and D for the second.

Wave/Pulse parameter w or p
     The default waveform is a pulse.

Frequency f
     C-Graph assumes a default frequency of 1Hz for values of f entered outside the range [0.5, N/4].

Scaling coefficient sc
     The scaling coefficient sc may be positive or negative. The maximum absolute value of sc for signals a, b, f, and h is N,while that for signals c,d,e, and g is 1. All signals will be scaled to unity for input values of sc outside the permitted range.

     With the default scaling coefficient of 1, signals a, b, f, and h are unit functions; signals d (square) an e (sawtooth) have a maximum amplitude of half the period (1/(2f)) while that of c (triangle) is one-quarter the period (1/(4f)).

Frequency Selection
===================

We can express the period P of a periodic signal as

     P   = N/(number of cycles)
       = T/(N/n)

where T is the duration ofthe signal register in seconds, N is the number of samples in the register (window length), and n is the number of samples in 1 period.

The frequency of the signal f is the reciprocal of the period, so

     f   = N/(nT) samples/seconds or Hz

C=Graph assumes that the duration of the signal register is 1 second, so

     f   = N/n) Hz

The sampling rate f_s is given by

     f_s   = N/T Hz

The interval h between successive samples being the reciprocal

     h   = 1/f_s seconds

If the window length and frequency chosen are 512 and 20 Hz (approximately the lower limit of the human audible range) then the number of samples n in each period would therefore be

     n   = N/f
       = 512/20
       = 25.6

C-Graph requires that n be a multiple of 4. For each periodic signal, the frequency entered by the user is accordingly adjusted so that n approximates to the nearest multiple of 4. So the frequency of the signal used by C-Graph would become

     f   = N/n
       = 512/26
       = 19.7 Hz

A C-Graph Session
*****************

In this session, we run C-Graph twice to compare the convolution of 2 signals of equal length:

 1. When the signals are finite sequences and the remaining interval is zero padded to the N with at least the same number of zeros as samples in each signal;

 2. When both signals extend across the full register of N samples.

We use a sawtooth pulse for the first signal, and a rectangular pulse half the amplitude of the sawtooth for the second signal. We also demonstrate the use of default values for unexpected input. The keychord ALT-TAB is used to toggle the terminal and the Gnuplot window.

In X, type c-graph. The splash screen will appear for a few seconds. Pressing ESC will kill the display, but you may invoke C-Graph without the splash screen with the --no-splash option. When the splash screen disappears, the following text will appear:

     GNU C-Graph version 2.0

     Dedicated to Eliezer Regnier and all victims of apartheid.'

     Copyright © 1982, 1983, 1996, 2008, 2009, 2010, 2011 Adrienne
     Gaye Thompson, Sole Author. GNU C-Graph version 2.0. Derived from
     BSc. dissertation "Interactive Computer Package Demonstrating: Sampling
     Convolution and the FFT", University of Aberdeen, Scotland (1983). For
     the code from the dissertation, visit
     <http://codeartnow.com/law-project>.

     GNU C-Graph is free software licensed under the terms of the GNU General
     Public License (the GPL) version 3 or later. You are welcome to
     distribute it under certain conditions.

     GNU C-Graph is distributed WITHOUT ANY WARRANTY
     without even the implied warranty of MERCHANTABILITY
     or FITNESS FOR A PARTICULAR PURPOSE.

     See the GPL section 15 regarding disclaimer of warranty.

     Press <g> then <ENTER> to display the GPL, or just press <ENTER> to
     continue.
     C-Graph:>> RET

     THIS IS GNU C-Graph - a tool for visualizing convolution.

     Compare the linear convolution of two signals with their circular
     convolution.

     Signal             Code
     ======             ====
     Sine               A
     Cosine             B
     Triangle           C
     Square             D
     Sawtooth           E
     Unit-Exponential   F
     Unit-Ramp          G
     Unit-Step          H

     Generate 2 signals from the above menu with up to 5 parameters:

     1. Number of samples N
     2. Signal code A to H
     3. Whether the signal, ifperiodic, is a wave w or a pulse p
     4. The frequency f
     5. The scaling coefficient sc

     Choose a value for "N" between 64 and 1024.
     C-Graph:>> 51

     The number of samples "N" is: 512

     Signal             Code
     ======             ====
     Sine               A
     Cosine             B
     Triangle           C
     Square             D
     Sawtooth           E
     Unit-Exponential   F
     Unit-Ramp          G
     Unit-Step          H

     Enter code for first signal
     C-Graph:>> e

     Is the signal periodic or is it a pulse?
     Type "w" for periodic wave, or "p" for pulse
     C-Graph:>> p

     Select the frequency "f" of this signal.
     C-Graph:>> 1.0

     The frequency of this signal is 1.00 Hz.

     Do you wish to scale this signal?
     Enter a value for the scaling coefficient "sc".
     A coefficient of 1 will give the unit function.
     C-Graph:>> 1

     Signal             Code
     ======             ====
     Sine               A
     Cosine             B
     Triangle           C
     Square             D
     Sawtooth           E
     Unit-Exponential   F
     Unit-Ramp          G
     Unit-Step          H

     Enter code for second signal
     C-Graph:>> s

     Is the signal periodic or is it a pulse?
     Type "w" for periodic wave, or "p" for pulse
     C-Graph:>> p

     Select the frequency "f" of this signal.
     C-Graph:>> 1

     The frequency of this signal is 1.00 Hz.

     Do you wish to scale this signal?
     Enter a value for the scaling coefficient "sc".
     A coefficient of 1 will give the unit function.
     C-Graph:>> .5

     You selected a square signal by default.

     Press <Enter> to see the signals in the time domain:>> RET

     ALT-TAB

     Hit <Enter> to continue:>> RET

     View the frequency-domain representation of the signals.
     Press <Enter> to see their FFTs:>> RET

     ALT-TAB

     Hit <Enter> to continue:>> RET

     Press <Enter> to compare linear and circular convolution:>> RET

     ALT-TAB

     Hit <Enter> to continue:>> RET

     Exiting GNU C-Graph ...
     Bye.

     galactica@regnier:~$ c-graph --no-splash

     GNU C-Graph version 2.0

     Dedicated to Eliezer Regnier and all victims of apartheid.'

     Copyright © 1982, 1983, 1996, 2008, 2009, 2010, 2011 Adrienne
     Gaye Thompson, Sole Author. GNU C-Graph version 2.0. Derived from
     BSc. dissertation "Interactive Computer Package Demonstrating: Sampling
     Convolution and the FFT", University of Aberdeen, Scotland (1983). For
     the code from the dissertation, visit
     <http://codeartnow.com/law-project>.

     GNU C-Graph is free software licensed under the terms of the GNU General
     Public License (the GPL) version 3 or later. You are welcome to
     distribute it under certain conditions.

     GNU C-Graph is distributed WITHOUT ANY WARRANTY
     without even the implied warranty of MERCHANTABILITY
     or FITNESS FOR A PARTICULAR PURPOSE.

     See the GPL section 15 regarding disclaimer of warranty.

     Press <g> then <ENTER> to display the GPL, or just press <ENTER> to
     continue.
     C-Graph:>> RET

     THIS IS GNU C-Graph - a tool for visualizing convolution.

     Compare the linear convolution of two signals with their circular
     convolution.

     Signal             Code
     ======             ====
     Sine               A
     Cosine             B
     Triangle           C
     Square             D
     Sawtooth           E
     Unit-Exponential   F
     Unit-Ramp          G
     Unit-Step          H

     Generate 2 signals from the above menu with up to 5 parameters:

     1. Number of samples N
     2. Signal code A to H
     3. Whether the signal, ifperiodic, is a wave w or a pulse p
     4. The frequency f
     5. The scaling coefficient sc

     Choose a value for "N" between 64 and 1024.
     C-Graph:>> 521

     The number of samples "N" is: 512.

     Signal             Code
     ======             ====
     Sine               A
     Cosine             B
     Triangle           C
     Square             D
     Sawtooth           E
     Unit-Exponential   F
     Unit-Ramp          G
     Unit-Step          H

     Enter code for first signal
     C-Graph:>> g

     Do you wish to scale this signal?
     Enter a value for the scaling coefficient "sc".
     A coefficient of 1 will give the unit function.
     C-Graph:>> 1

     Signal             Code
     ======             ====
     Sine               A
     Cosine             B
     Triangle           C
     Square             D
     Sawtooth           E
     Unit-Exponential   F
     Unit-Ramp          G
     Unit-Step          H

     Enter code for second signal
     C-Graph:>> h

     Do you wish to scale this signal?
     Enter a value for the scaling coefficient "sc".
     A coefficient of 1 will give the unit function.
     C-Graph:>> q56

     That was not a number. Try again!
     C-Graph:>> 256

     Press <Enter> to see the signals in the time domain:>> RET

     ALT-TAB

     Hit <Enter> to continue:>> RET

     View the frequency-domain representation of the signals.
     Press <Enter> to see their FFTs:>> RET

     ALT-TAB

     Hit <Enter> to continue:>> RET

     Press <Enter> to compare linear and circular convolution:>> RET

     ALT-TAB

     Hit <Enter> to continue:>> RET

     Exiting GNU C-Graph ...
     Bye.

On exit, the directory c-graphs will be created. Directory c-graphs will have 2 subdirectories containing the graphs, Gnuplot command files, and the data used for plotting generated by the last run. The subdirectories and files are:

  * c-graphs/graphs: signals.png, transforms.png, convolutions.png

  * c-graphs/coms: signals.cg, transforms.cg, convolutions.cg time.dat trans.dat

One can then print the graphs and display them either by using a graphics editor like Image Magick, or by executing the command files with Gnuplot.

Reporting Bugs
**************

To report bugs or suggest enhancements for GNU C-Graph, please send electronic mail to agt@codeartnow.com.

For bug reports, please include enough information for the maintainers to reproduce the problem.  Generally speaking, that means:

  * The version numbers of GNU C-Graph (which you can find by running c-graph --version) and any other program(s) or manual(s) involved.

  * Hardware and operating system names and versions.

  * The contents of any input files necessary to reproduce the bug.

  * The expected behavior and/or output.

  * A description of the problem and samples of any erroneous output.

  * Options you gave to configure other than specifying installation directories.

  * Anything else that you think would be helpful.

When in doubt whether something is needed or not, include it.  It's better to include too much than to leave out something important.

Patches are welcome. Please follow the existing coding style.

Sketch of Convolution Theory
****************************

Introductory Ideas
==================

GNU C-Graph compares the linear and circular convolution of two signals. Subroutine convo computes the linear convolution directly in the time domain, while the FFT is exploited to compute circular convolution through the convolution theorem, which is defined below (see The Convolution Theorem).

Convolution is an operation by which two functions combine to produce a third that represents a kind of moving average. This is a naturally ocurring phenomenon that presents itself whenever there is a linear system obeying the principles of superposition and shift/time invariance. Accordingly, the mathematics of convolution has found application to much of science and engineering in areas ranging from statistics to computer vision.

The output of any linear shift invariant system may be described as the convolution of the input with the impulse response of the system. In computer vision, for example, where the system being considered is a 2-dimensional image, the output of the system may be blurred as a result of the relative motion of the camera and the object. This blurred image can be modelled by convolution of the static image with the 2-dimensional impulse response.

The 2-dimensional impulse response is called a pointspread function (PSF). Each pixel in the image produces a copy of the PSF, scaled according to the strength of the pixel and spatially shifted. Superposition of these copies form the resultant output signal, the system being linear and shift invariant.  The output blurred image is then a convolution that is, in fact, a linear combination of the PSFs. The design of a filter for image restoration must then rely on inverse convolution.

A thorough treatment of the mathematics of convolution is beyond the scope of this manual. See Appendix B, for some references on the subject, and related engineering theory.

The Mathematics
===============

Consideration of the 1-dimensional case simplifies the arithmetic. To prove the convolution theorem, we first derive an expression for the convolution of 2 signals, then apply the Fourier transform to this expansion.

Deriving the Convolution Sum
----------------------------

A discrete-time signal may be modelled as a series of piecewise rectangular pulses. The summation of all such rectangular pulses approximates the signal f:

     f(n)   = \sum_m f(m) rect(n - m)

where n - m denotes the rectangle whose base on the n axis is centred at sample n=m.

In the limit, the series of rectangular pulses approaches a continuous signal as the pulse width tends to zero and each pulse becomes an impulse signal. Each impulse signal can then be represented as a scaled and shifted unit impulse simulating one sample of the discrete signal.

     f(n)   = \sum_m f(m) \delta(n - m)

Applying a system transform M that maps the input signal f to the output signal g,

     g(n)   = M[f(n)]
     g(n)   = M [f(n) = \sum_m f(m) \delta(n - m)]
       = \sum_m f(m) M[\delta(n - m)]

Since the system transforms a delta function to the system impulse response h

     ___

     \
     g[n] =
     /__ f[m] h[n - m]     (1)

       m

The above expression called the convolution sum, denoted by f(*)h, defines the output g(n) of the system.

The Convolution Theorem
-----------------------

GNU C-Graph demonstrates the convolution theorem. The convolution of 2 signals in the time domain is equal to the inverse Fourier transform of the product of their transforms in the frequency domain.

Just as a signal can be represented by a linear combination of scaled and shifted impulses, we can also describe the signal as a linear combination of sinusoidal basis functions. The Fourier transform exploits this representation to deconstruct the signal into frequency components, each corresponding to a basis sinusoid.

Using Euler's identity

     e^{j\theta}   = cos(\theta) + j \sin(\theta)

The sinusoidal sum representing the discrete time signal may be written in the form

     f(n)   = 1/N \sum\limits_{k=0}^{N-1} F(k) e^{j\omega_kn}

where \omega_k = 2k\pi/N, and the F(k) are Fourier transform coefficients indicating the strength of the kth spectral sample of frequency \omega_k (how much of the each basis sinusoid is present in the signal). (For a periodic signal, these coefficients are the \delta functions of the Fourier transform.)

Accordingly, the F(k) may be computed from the signal

     F(k)
     = \sum\limits_{n=0}^{N-1} f(n) e^{-j\omega_kn}

This is the Fourier transform description of the signal as a function of frequency.

From eqn (1), the convolution of an input signal f with the system impulse response h to give an output g is defined as:

     g(n)   = f(*)h = \sum_m f(m) h(n - m)

Let the Fourier transform of g(n) be denoted by

     \Gamma[g(n)]   = g(n) e^{-j{\omega}n}, then
     \Gamma[f(*)g]   = \sum_n [f(*)g] e^{-j{\omega}n}
     \Gamma[f(*)h]
     = \sum_n \sum_m f(m) h(n - m)] e^{-j{\omega}n}
       = \sum_m f(m) \sum_n h(n - m) e^{-j{\omega}n}

Changing the variable to p = n - m

     \Gamma[g(n)]
     = \sum_m f(m) \sum_p h(p) e^{-j{\omega}(m+p)}
       = \sum_m f(m)e^{-jm\omega} \sum_p h(p)e^{-j{\omega}p}

Taking the inverse Fourier transform,

     f(*)g   = \Gamma^{-1}[ \Gamma[f] \Gamma[h]

This is the convolution theorem.

Linear and Circular Convolution
===============================

The simulation of actual or linear convolution requires a sequence of multiplications and additions that are computationally too slow for high speed operations such as deblurring filters for precision robotic vision control systems. The FFT, an algorithm for efficiently computing the DFT, dramatically overcomes the computational load by successively decomposing the multiplication of two sequences into subsequences of half the length thereby reducing the number of artithmetic operations by roughly N/logN.

The cost of this additional computational power is the treatment of the convolving signals as periodic with N samples per period. The resulting convolution is termed circular convolution. It can be shown that circular convolution and linear convolution are equivalent if N \ge L + P -1 where L, M are the unpadded lengths of the sequences being convolved.

We illustrate the difference between linear and circular convolution using abbreviated sequences for the pulses demonstrated in A C-Graph Session

Linear Convolution
------------------

As noted above (see Introductory Ideas), in linear convolution, each sample of f contributes a scaled and shifted copy of the h. This is accomplished by the multiplication of the particular sample m of f by each sample of h.

This sequential multiplication can be visualized as a physical reflection of h(m)) about the vertical axis to obtain h(-m) followed by discrete shifts of 1 sample interval (\delta n = 1) along the time axis with no overlap of the signals at the beginning and end of the translation. As the impulse response moves along the time axis, the point by point multiplication of coincident samples is summed. The sum at each point in the translation is the value of the convolution sum g(n) at that point, and the length of the convolution is L + M.

For the sequences

     f(m)   = [1,1,1], and
     h(m)   = [0,1,2]

The series of operations for linear convolution

     f(*)h    = [0 1 3 3 2], are:

     0.        1 1 1
        2 1 0
        ---------
     g(0) = f(m)h(0-m)
      0+0+0+0+0 = 0

     1.        1 1 1
          2 1 0
          -------
     g(1) = f(m)h(1-m)
        0+1+0+0 = 1

     2.        1 1 1
            2 1 0
            -----
     g(2) = f(m)h(2-m) =
          2+1+0 = 3

     3.        1 1 1
              2 1 0
            -------
     g(3) = f(m)h(3-m) =
          0+2+1+0 = 3

     4.        1 1 1
                2 1 0
            ---------
     g(4) = f(m)h(4-m) =
          0+0+2+0+0 = 2

Circular Convolution
--------------------

We can imagine circular convolution in terms of the relative rotation of two concentric cylinders whose circumferences are of length N. A copy of the N samples comprising f is wrapped anticlockwise round one cylinder, while a copy of h is wrapped clockwise round the other cylinder, reflecting h . Rotating the second cylinder anticlockwise by 1 sample interval each time, multiplying the coincident samples and summing will give corresponding values of the convolved signal g.

     f(m)   = [1,1,1], and
     h(m)   = [0,1,2]

The series of operations for circular convolution

     f(*)h   = [0 1 3 3 2], are:

     0.    1 1 1
        0 2 1
        -----
     g(0) = f(m)h(0-m)
      0+2+1 = 3

     1.    1 1 1
        1 0 2
        -----
     g(1) = f(m)h(1-m)
      1+0+2 = 3

     2.    1 1 1
        2 1 0
        -----
     g(2) = f(m)h(2-m)
      2+1+0 = 3

By zero-padding each sequence of length L = 3 to length N so that N\ge L + L -1 (see Linear and Circular Convolution), we obtain the sequences

     f(m)   = [1,1,1,0,0]
     h(m)   = [0,1,2,0,0]

Circular convolution then achieves the same as result as linear convolution:

     f(*)h   = [0 1 3 3 2]

The operations are:

     0.    1 1 1 0 0
        0 0 0 2 1
        ---------
     g(0) = f(m)h(0-m)
      0+0+0+2+1 = 0

     1.    1 1 1 0 0
        1 0 0 0 2
        -----
     g(1) = f(m)h(1-m)
      1+0+0+0+2 = 1

     2.    1 1 1 0 0
        2 1 0 0 0
        ---------
     g(2) = f(m)h(2-m)
      2+1+0+0+0 = 3

     3.    1 1 1 0 0
        0 2 1 0 0
        ---------
     g(3) = f(m)h(3-m)
      0+2+1+0+0 = 3

     4.    1 1 1 0 0
        0 0 2 1 0
        ---------
     g(4) = f(m)h(4-m)
      0+0+2+1+0 = 2

References
**********

The sources below were consulted in the preparation of GNU C-Graph and/or the 1983 dissertation [1] from which GNU C-Graph is derived.

 1. Thompson, Adrienne G. ''Interactive Computer Package Demonstrating: Sampling Convolution and the FFT'', BSc. Engineering Honours thesis, University of Aberdeen (Scotland), 1983, see http://codeartnow.com/law-project.

 2. Horn, Berthold, K. P. Robot Vision. MIT Press, Cambridge, Massachusetts, 1986.

 3. McGillem, Clare D., and Cooper, George R. Continuous and Discrete Signal and System Analysis. Holt, Rinehart and Winston, Inc., 1990.

 4. Oppenheim, Alan V., and Schafer, Ronald W. Digital Signal Processing. Prentice-Hall, Englewood Cliffs, N.J., 1975.

 5. Smith, Steven W. The Scientist and Engineer's Guide to Digital Signal Processing http://www.dspguide.com

 6. Stremler, Ferrel G. Introduction to Communication Systems. Addison-Wesley Publishing Co. Inc, 1977.

 7. Wouk, Arthur. fft-simple.f90. Ed. Alan Miller. 2003. Retrieved from http://jblevins.org/mirror/amiller/fft_simple.f90

 8. Yuen, C. K. and Fraser, D. Digital Spectral Analysis. CSIRO/Pitman, East Melbourne, Australia, 1979.

 9. Ziemer, Rodger E., Tranter, William H., Fannin, D. Ronald. Signals Systems Continuous and Discrete. 4th ed. Prentice Hall, Upper Saddle River, NJ 07458 1998.

GNU Free Documentation License
******************************

     [file fdl.texi not included]