NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Numerical Tools in Matlab

In Matlab, root-finding is always numerical:^2.3

>> % polynomial = array of coefficients in Matlab
>> p = [1 0 0 0 5 7]; %  p(x) = x^5 + 5*x + 7
>> format long;       %  print double-precision
>> roots(p)           %  print out the roots of p(x)
ans =
1.30051917307206 + 1.10944723819596i
1.30051917307206 - 1.10944723819596i
-0.75504792501755 + 1.27501061923774i
-0.75504792501755 - 1.27501061923774i
-1.09094249610903

Matlab has the following primitives for complex numbers:

>> help j

J      Imaginary unit.
The variables i and j both initially have the value sqrt(-1)
for use in forming complex quantities.  For example, the
expressions 3+2i, 3+2i, 3+2i, 3+2j and 3+2*sqrt(-1).
all have the same value.  However, both i and j may be
assigned other values, often in FOR loops and as subscripts.
    See also I.
    Built-in function.
    Copyright (c) 1984-92 by The MathWorks, Inc.
>> sqrt(-1)
ans =
0 + 1.0000i
>> help real
REAL Complex real part.
REAL(X) is the real part of X.
    See also IMAG, CONJ, ANGLE, ABS.
>> help imag
IMAG Complex imaginary part.
IMAG(X) is the imaginary part of X.
See I or J to enter complex numbers.
    See also REAL, CONJ, ANGLE, ABS.
>> help conj
CONJ Complex conjugate.
CONJ(X) is the complex conjugate of X.
>> help abs
ABS Absolute value and string to numeric conversion.
ABS(X) is the absolute value of the elements of X. When
X is complex, ABS(X) is the complex modulus (magnitude) of
the elements of X.
    See also ANGLE, UNWRAP.

    ABS(S), where S is a MATLAB string variable, returns the
    numeric values of the ASCII characters in the string.
    It does not change the internal representation, only the
    way it prints.
    See also SETSTR.
>> help angle
ANGLE Phase angle.
ANGLE(H) returns the phase angles, in radians, of a matrix with
complex elements.
    See also ABS, UNWRAP.</pre></p><p>Note how helpful the ''See also'' information is in Matlab.</p><p>Let's run through a few elementary manipulations of complex
numbers in Matlab:
>> x = 1; % Every symbol must have a value in Matlab
>> y = 2;
>> z = x + j * y

z =
1.0000 + 2.0000i
>> 1/z
ans =
0.2000 - 0.4000i
>> z^2
ans =
-3.0000 + 4.0000i
>> conj(z)
ans =
1.0000 - 2.0000i
>> z*conj(z)
ans =
5
>> abs(z)^2
ans =
5.0000
>> norm(z)^2
ans =
5.0000
>> angle(z)
ans =
1.1071
Now let’s do polar form:
>> r = abs(z)

r =
2.2361
>> theta = angle(z)
theta =
1.1071
>> r * exp(j * theta)
ans =
1.0000 + 2.0000i
>> z
z =
1.0000 + 2.0000i
>> z/abs(z)
ans =
0.4472 + 0.8944i
>> exp(j*theta)
ans =
0.4472 + 0.8944i
>> z/conj(z)
ans =
-0.6000 + 0.8000i
>> exp(2jtheta)
ans =
-0.6000 + 0.8000i
>> imag(log(z/abs(z)))
ans =
1.1071
>> theta
theta =
1.1071
>>
Some manipulations involving two complex numbers:
>> x1 = 1;
>> x2 = 2;
>> y1 = 3;
>> y2 = 4;
>> z1 = x1 + j * y1;
>> z2 = x2 + j * y2;
>> z1

z1 =
1.0000 + 3.0000i
>> z2
z2 =
2.0000 + 4.0000i
>> z1*z2
ans =
-10.0000 +10.0000i
>> z1/z2
ans =
0.7000 + 0.1000i
Another thing to note about Matlab is that the transpose operator '
(for vectors and matrices) conjugates as well as transposes. Use .'
to transpose without conjugation:
>>x = [1:4]*j

x =
0 + 1.0000i        0 + 2.0000i        0 + 3.0000i        0 + 4.0000i
>> x'
ans =
0 - 1.0000i
0 - 2.0000i
0 - 3.0000i
0 - 4.0000i
>> x.'
ans =
0 + 1.0000i
0 + 2.0000i
0 + 3.0000i
0 + 4.0000i
>>
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