Complex Number Manipulation

Let's run through a few elementary manipulations of complex numbers in Matlab:

>> x = 1; 
>> y = 2;
>> z = x + j * y

z =
   1 + 2i

>> 1/z

ans =
   0.2 - 0.4i

>> z^2

ans =
  -3 + 4i

>> conj(z)

ans =
   1 - 2i

>> z*conj(z)

ans =
     5

>> abs(z)^2

ans =
    5

>> norm(z)^2

ans =
    5

>> angle(z)

ans =
    1.1071

Now let's do polar form:

>> r = abs(z)

r =
    2.2361

>> theta = angle(z)

theta =
    1.1071

Curiously, is not defined by default in Matlab (though it is in Octave). It can easily be computed in Matlab as e=exp(1).

Below are some examples involving imaginary exponentials:

>> r * exp(j * theta)

ans =
   1 + 2i

>> z

z =
   1 + 2i

>> z/abs(z)

ans =
   0.4472 + 0.8944i

>> exp(j*theta)

ans =
   0.4472 + 0.8944i

>> z/conj(z)

ans =
  -0.6 + 0.8i

>> exp(2*j*theta)

ans =
  -0.6 + 0.8i

>> imag(log(z/abs(z)))

ans =
    1.1071

>> theta

theta =
    1.1071

>>

Here are 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 + 3i

>> z2

z2 =
   2 + 4i

>> z1*z2

ans =
 -10 +10i

>> z1/z2

ans =
   0.7 + 0.1i

Another thing to note about matlab syntax 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 + 1i   0 + 2i   0 + 3i   0 + 4i

>> x'

ans =
        0 - 1i
        0 - 2i
        0 - 3i
        0 - 4i

>> x.'

ans =
        0 + 1i
        0 + 2i
        0 + 3i
        0 + 4i

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.