the complex exponential (or two-sided) fourier series general information and example 2.5
TRANSCRIPT
The complex exponential (or two-sided) Fourier seriesGeneral information and Example 2.5
Two-Sided or Complex Exponential Form of the Fourier Series
Advantages: Treats dc term the same as all other terms Complex form is used to derive Fourier transform for
nonperiodic waveforms (discussed later) Complex form is also the basis for the discrete Fourier
transform (continued)
x t c e
cT
x t e dt
nn
j nf t
nj nf t
t
t T
o
o
o
o
( )
( )
2
21
where
Two-Sided or Complex Exponential Form (continued)
Disadvantages: Form is not intuitive due to
Use of complex exponentialsComplex cn coefficientsResultant “negative frequency” components
(continued)
x t c e
cT
x t e dt
nn
j nf t
nj nf t
t
t T
o
o
o
o
( )
( )
2
21
where
Plotting cn in the Complex Plane
Since Euler’s identity produces a real cosine term and an imaginary sine term, the cn coefficients of the two-sided Fourier series are also complex numbers. We can plot such a number in the complex plane (real portion represented by the x axis, imaginary portion represented by the y axis).
x t c e
cT
x t e dt
nn
j nf t
nj nf t
t
t T
o
o
o
o
( )
( )
2
21
where
Plotting cn in the Complex Plane
n
cn
Imaginary axis
Real axis
Im{cn}
Re{cn}
We can express cn in terms of a real and imaginary component, or in terms of a magnitude |cn| and a phase n.
From our earlier work, we know how to relate cn and n toreal-world quantities, the phase and magnitude of the one-sided form:
for n = 0, cn = Xn for n > 0, cn =0.5 Xn and phase = nfor n < 0, cn =0.5 X-n and phase = -n
Example 2.5Use the complex exponential form of the Fourier series to represent the signal x(t) shown in Examples 2.3 and 2.4 (reproduced below). Draw the two-sided magnitude and phase spectra of the signal.
0-5-10 5 10 seconds
volts
1
2
3x(t)
Solution to Example 2.5In Example 2.4 we determined the Xn and n coefficients of the one-sided form of the Fourier series for x(t). As shown earlier,
for n = 0, cn = Xn for n > 0, cn =0.5 Xn and phase = nfor n < 0, cn =0.5 X-n and phase = -n
The two-sided magnitude and phase spectra are thus plotted below:
180
135
90
45
-45
-90
-135
-180
Phase indegrees
0.6
0.5
0.4
0.3
0.2
0.1
Frequency in Hz
Magnitudein volts
-3 -2 -1 1 32
Frequency in Hz
Interpreting the “Negative Frequency” Components Produced by the Two-Sided Fourier Series
The magnitude and phase spectra produced by the one-sided form of the Fourier series have physical meaning.
The magnitude and phase spectra produced by the two-sided form of the Fourier series do not have physical meaning per se.
“Negative frequency” does not exist in the real world — it is just a mathematical concept needed to correlate the one-sided and two-sided forms of the Fourier series.
The “negative frequency” components of the two-sided Fourier series (corresponding to the summation from n = - to n = -1) physically represent additional contributions at the corresponding positive frequency.
(continued)
Interpreting the “Negative Frequency” Components ... (continued)
When determining real-world magnitude or power, you must therefore consider (i.e., add) both the positive and corresponding “negative” frequency components. You can ignore the phase of the two-sided “negative frequency” components.
Channel with 1Hz bandwidth passes first five harmonics
sec-2 -1 1 2 3 4 5 6 7 8-0.5
0.5
1
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2.5 volts
Channel with1Hz bandwidth
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Frequency in Hz
Magnitudein volts
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Frequency in Hz
Magnitudein volts
sec-2 -1 1 2 3 4 5 6 7 8-0.5
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2.5 volts
Channel with 2Hz bandwidth passes first ten harmonics
Channel with2Hz bandwidth
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Frequency in Hz
Magnitudein volts
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Frequency in Hz
Magnitudein volts
sec-2 -1 1 2 3 4 5 6 7 8-0.5
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sec-2 -1 1 2 3 4 5 6 7 8-0.5
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Channel with 3Hz bandwidth passes first fifteen harmonics
Channel with3Hz bandwidth
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Frequency in Hz
Magnitudein volts
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Frequency in Hz
Magnitudein volts
sec-2 -1 1 2 3 4 5 6 7 8-0.5
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2.5 volts
sec-2 -1 1 2 3 4 5 6 7 8-0.5
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2.5 volts