Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 776

B.1.2 Amplitude-Phase Form
From the sine-cosine form, we notice that there is a sum of two terms with the same frequency. The
term in the first sum is a n cos ðnu 0 tÞ while the other is b n sin ðnu 0 tÞ . We can combine these two terms
and modify the sine-cosine form into the amplitude-phase form:
x t ¼ A 0 þ

N
n¼ 1 A n cos ðnu 0 t þ f n Þ

(B.5)

The DC term is same as before, that is,

A 0 ¼ a 0 (B.6)

and the amplitude and phase are given by

q

a

2 2

n

A n ¼

n þ b

(B.7)

f n ¼ tan 1 b n
a n

(B.8)

respectively. The amplitude-phase form provides very useful information for spectral analysis. With
the calculated amplitude and phase for each harmonic frequency, we can create the spectral plots. One
depicts a plot of the amplitude versus its corresponding harmonic frequency (the amplitude spectrum),
while the other plot shows each phase versus its harmonic frequency (the phase spectrum). Note that
the spectral plots are one-sided, since amplitudes and phases are plotted versus the positive harmonic
frequencies. We will illustrate these in Example B.1.

B.1.3 Complex Exponential Form
The complex exponential form is developed based on expanding sine and cosine functions in the sine-
cosine form into their exponential expressions using Euler's formula and regrouping these exponential
terms. Euler's formula is given by

e jx ¼ cos x j sin x

which can be written as two separate forms:
cos x ¼ e jx þ e jx
2

sin x ¼ e jx e jx
2 j

We will focus on interpretation and application rather than the derivation of this form. Thus the
complex exponential form is expressed as

N

x t ¼ c n e jnu 0 t (B.9)

n¼ N