Z Rocks!!

First, the background:

Euler's Identity: e^j&omega = cos(&omega) + j sin(&omega)

where j = sqrt(-1) and ln(e) = 1

Some cool consequences of Euler's Identity:

• e^j&pi = -1
• e^j&pi/2 = j
• e^-j&pi/2 = -j
• e^j&pi/4 = 0.707 + 0.707j

And now... The Star of Our Show:

Z = e^j&omega

There is much we can do with Z. For example:

Making discretely sampled sine waves from Z:

let &omega = &pi/4

Successively raize Z to higher powers:
(successively multiply the prior sample result by Z):

Z0	=	1.0
Z1	= ej&pi/4 =	0.707 + 0.707j
Z2	= ej&pi/2 =	j
Z3	=	-0.707 + 0.707j
Z4	=	-1
Z5	=	-0.707 + -0.707j
Z6 =	=	-j
Z7 =	=	0.707 - 0.707j
Z8 =	=	1.0
etc.	etc.	etc.

So this complex "Phasor" generates a real cosine wave
and an imaginary sine wave at 1/8 the sample rate.
This is also called a "quadrature oscillator."

We can get the cosine by taking the real part:

cos(&omega t) = Re[e^{&omega t}]

and the sine by taking the imaginary part:

sin(&omega t) = Im[e^{&omega t}]

What else can we do with Z?

The Z Transform!

Given a difference equation of the form:

y(n) = x(n) + a₁x(n-1) + a₂x(n-2) + ... + a_Mx(n-M)
                      - b₁y(n-1) - b₂y(n-2) - ... - b_My(n-M)

The Z transform of this is:

Y(&omega) = X + a₁XZ^-1 + a₂XZ^-2 + ... + a _MXZ^-M
              - b₁YZ^-1 - b₂YZ^-2 - ... - b_MYZ ^-M

Some algebra yields the "Transfer Function":

Y/X (&omega) = H(&omega) =

(1 + a₁Z^-1 + a₂Z^-2 + ... + a_MZ ^-M)
----------------------------------------------
(1 + b₁Z^-1 + b₂Z^-2 + ... + b_MZ^-M)

If we want to know the magnitude frequency response of
a filter at any frequency, then we just need to plug in for
&omega, collect real terms and imaginary terms, and compute
the squareroot of the sum of the squared real and squared
imaginary terms. The magnitude of a ratio is the ratio of
the magnitudes (| numerator | / | denominator |), so

| H | = Sqrt[ Re(numerator)² + Im(numerator)² ] /
                     
Sqrt[ Re(denominator)² + Im(denominator)² ]

If we want the phase, we compute Arctan (Im/Re).
The phase angle of a ratio of phasors is the numerator
angle minus the denominator angle:

Arctan (Im(num.) / Re(num.)) - Arctan (Im(denom.) / Re(denom.))

Let's do a specific magnitude example:

y(n) = x(n) + y(n-1)

Y(&omega) = X(&omega) + Y(&omega)Z^-1

H(&omega) = Y/X(&omega) = 1 / (1 - cos(&omega) + jsin(&omega))

| H (&omega) | = 1 / sqrt[1-2cos(&omega)+cos²(&omega) + sin²(&o mega)]
= 1 / sqrt(2-2cos(&omega)) = 1 / sqrt(sin²(2&omega)) = 1/sin(2&omega)

YAY!!