Communication
Systems
Fourier Series
By
Engr. Jawwad Ahmad (Ph.D)
1
Today’s Goal
Fundamental Frequency & Harmonics
Composite Signals
Fourier Series
Engr. Dr. Jawwad Ahmad 2
Fundamental Frequency,
Harmonics & Composite Signal
Any non-sinusoidal periodic waveform can be expressed as a
very large number of sinusoidal waveform of different amplitudes
and frequencies such that the frequencies of sinusoidal are integral
multiple of one lowest frequency called Fundamental Frequency.
Any well-behaved periodic waveform can be represented as a
series of sine and/or cosine waves at multiples of its fundamental
frequency plus (sometimes) a dc offset. This is known as a Fourier
series.
Engr. Dr. Jawwad Ahmad 3
Fundamental Frequency,
Harmonics & Composite Signal
This very useful (and perhaps rather surprising) fact was
discovered in 1822 by Joseph Fourier, a French mathematician,
in the course of research on heat conduction.
Fourier's discovery, applied to a time-varying signal, can be
expressed mathematically as follows:
Engr. Dr. Jawwad Ahmad 4
Fourier Series
where
f(t) = any well-behaved function of time as described
above. For our purposes, f(t) will generally be either a
voltage v(t) or a current i(t).
An & Bn = real-number coefficients; that is, they can be
positive, negative, or zero.
ω = radian frequency of the fundamental.
Engr. Dr. Jawwad Ahmad 5
Fourier Series
Consider a signal of period = 1, frequency = 1, time period = 1.
sin 2 t
sin 2 (1)t
Another periodic signal having period = 1, frequency = 2, time period = ½.
sin 4 t
sin 2 (2)t
Another periodic signal having period = 1, frequency = 3, time period = 1/3.
sin 6 t
sin 2 (3)t
On combining these periodic functions
f (t ) sin 2 t sin 4 t sin 6 t
Engr. Dr. Jawwad Ahmad 6
Fourier Series
sin 2 (1)t
sin 2 (2)t
sin 2 (3)t
f (t ) sin 2 t sin 4 t sin 6 t 7
Fourier Series
To model a complicated signal, we can modify amplitude, frequency and phase,
so, N
f (t ) An sin( 2 f n t n )
n 1
From trigonometric function
An sin 2 f nt n An sin 2 f n t cos n An cos 2 f n t sin n
An sin n cos 2 f n t An cos n sin 2 f nt
Let, An sin And
n an An cos n bn
Therefore,
An sin 2f n t n an cos 2f nt bn sin 2f n t
Engr. Dr. Jawwad Ahmad 8
Fourier Series
Thus, N
f (t ) an cos 2 f n t bn sin 2 f n t
n 1
Adding a constant term normally known as DC-component (ao)
N
f (t ) a0 an cos 2 f n t bn sin 2 f n t
n 1
or,
N
f (t ) a0 an cos 2 nf1t bn sin 2 nf1t
n 1
Engr. Dr. Jawwad Ahmad 9
Fourier Series
In terms of Angular Frequency,
N
f (t ) a0 an cos n t bn sin n t
n 1
or,
N
f (t ) a0 an cos n1t bn sin n1t
n 1
where ao, an, bn are the constant co-efficient
for the corresponding frequency which
depends upon f(t).
Engr. Dr. Jawwad Ahmad 10
Fourier Series
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Fourier Series
12
Thank you
Engr. Dr. Jawwad Ahmad 13
