A well known difference equation exists which can compute samples of
parametrically well behaved sinusoids
[1,2]. The frequency of the sinusoid
can easily be changed without changing the amplitude. The equation is
a property of the multiplication of two complex numbers.^{1}Complex numbers can be represented as two dimensional vectors in a
plane in which the x axis is the real part of the number and the y
axis is the imaginary part of the number. The polar coordinate form of
the vector is a magnitude and an angle (measured from the x axis).
The magnitude of the product of two numbers is the product of the
magnitudes of the two numbers and the angle of the product is the sum
of the angles of the two numbers.

We will use the following definitions and relations:

In terms of real and imaginary components:

Let a sequence of complex numbers

where . The initial state is taken to be , i.e., the initial magnitude and angle are and , respectively. By the rules of complex number multiplication, we have

By changing , the period can be changed without affecting the magnitude.

The actual difference equations to compute the sine wave are obtained
by writing Eq.(1) as real and imaginary parts. This yields two
difference equations which can be extrapolated to compute
and :

where and .

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