Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Finding the Fundamental Frequency
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Often one is given a set of frequencies f1 , f2 , . . . , fN and is
required to find the fundamental frequency f0 .
Specifically, this means one must find a frequency f0 and
integers n1 , n2 , . . . , nN such that all of the following
equations are met:
f1
f2
= n 1 · f0
= n 2 · f0
..
.
fN
= nN · f0
Note that there isn’t always a solution to the above
problem.
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©2016, B.-P. Paris
However, if all frequencies are integers a solution exists.
Even if all frequencies are rational a solution exists.
ECE 201: Intro to Signal Analysis
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Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Example
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Find the fundamental frequency for the set of frequencies
f1 = 12, f2 = 27, f3 = 51.
Set up the equations:
12 = n1 · f0
27 = n2 · f0
51 = n3 · f0
Try the solution n1 = 1; this would imply f0 = 12. This
cannot satisfy the other two equations.
Try the solution n1 = 2; this would imply f0 = 6. This
cannot satisfy the other two equations.
Try the solution n1 = 3; this would imply f0 = 4. This
cannot satisfy the other two equations.
Try the solution n1 = 4; this would imply f0 = 3. This can
satisfy the other two equations with n2 = 9 and n3 = 17.
©2016, B.-P. Paris
ECE 201: Intro to Signal Analysis
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Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Example
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Note that the three sinusoids complete a cycle at the same
time at T0 = 1/f0 = 1/3s.
3
2
Amplitude
1
0
−1
f=12Hz
f=27Hz
f=51Hz
Sum
−2
−3
0
©2016, B.-P. Paris
0.05
0.1
0.15
0.2
Time (s)
0.25
0.3
0.35
ECE 201: Intro to Signal Analysis
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Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
A Few Things to Note
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Note that the fundamental frequency f0 that we determined
is the greatest common divisor (gcd) of the original
frequencies.
I f = 3 is the gcd of f = 12, f = 27, and f = 51.
0
1
2
3
The integers ni are the number of full periods (cycles) the
sinusoid of freqency fi completes in the fundamental period
T0 = 1/f0 .
I For example, n = f · T = f · 1/f = 4.
1
1
0
1
0
I The sinusoid of frequency f completes n = 4 cycles
1
1
during the period T0 .
©2016, B.-P. Paris
ECE 201: Intro to Signal Analysis
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Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Exercise
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Find the fundamental frequency for the set of frequencies
f1 = 2, f2 = 3.5, f3 = 5.
©2016, B.-P. Paris
ECE 201: Intro to Signal Analysis
108
Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Fourier Series
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We have shown that a sum of sinusoids with harmonic
frequencies is a periodic signal.
One can turn this statement around and arrive at a very
important result:
Any periodic signal can be expressed as a
sum of sinusoids with harmonic frequencies.
The resulting sum is called the Fourier Series of the signal.
Put differently, a periodic signal can always be written in
the form
x (t ) = A0 + ÂN
i =1 Ai cos(2pif0 t + fi )
j2pif0 t + X ⇤ e j2pif0 t
= X0 + ÂN
i = 1 Xi e
i
with X0 = A0 and Xi =
©2016, B.-P. Paris
Ai jfi
2e .
ECE 201: Intro to Signal Analysis
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Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Fourier Series
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For a periodic signal the complex amplitudes Xi can be
computed using a (relatively) simple formula.
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Specifically, for a periodic signal x (t ) with fundamental
period T0 the complex amplitudes Xi are given by:
1
Xi =
T0
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©2016, B.-P. Paris
Z T0
0
x (t ) · e
j2pit /T0
dt.
Note that the integral above can be evaluated over any
interval of length T0 .
ECE 201: Intro to Signal Analysis
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Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
Example: Square Wave
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A square wave signal is periodic and between t
t = T0 it equals
(
1
0  t < T20
x (t ) =
1 T20  t < T0
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From the Fourier Series expansion it follows that x (t ) can
be written as
x (t ) =
•
Â
n =0
©2016, B.-P. Paris
4
(2n
1) p
cos(2p (2n
1)ft
ECE 201: Intro to Signal Analysis
0 and
p/2)
111
Sum of Sinusoidal Signals
Time-Domain and Frequency-Domain
Periodic Signals
25-Term Approximation to Square Wave
x (t ) =
25
Â
n =0
4
(2n
1) p
cos(2p (2n
1)ft
p/2)
1.5
1
Amplitude
0.5
0
−0.5
−1
−1.5
0
©2016, B.-P. Paris
0.01
0.02
0.03
Time (s)
0.04
0.05
0.06
ECE 201: Intro to Signal Analysis
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