1. The complex exponential (or two-sided) Fourier series General information and Example 2.5

2. 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)

3. Two-Sided or Complex Exponential Form (continued)
Disadvantages:
Form is not intuitive due to
Use of complex exponentials
Complex cn coefficients
Resultant “negative frequency” components
(continued)

4. 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).

5. Plotting cn in the Complex Planefn 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 fn.
From our earlier work, we know how to relate cn and fn to
real-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 = fn
for n < 0, cn =0.5 X-n and phase = - f-n

6. Example 2.5
Use 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.
volts
x(t) 1 2 3
· · ·
· · · -5
-10 5 10
seconds

7. Solution to Example 2.5
In Example 2.4 we determined the Xn and fn 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 = fn
for n < 0, cn =0.5 X-n and phase = - f-n
The two-sided magnitude and phase spectra are thus plotted below:
0.6
0.5
0.4
0.3
0.2
0.1
Frequency in Hz
Magnitude
in volts
180
Phase in
degrees
135 90 45 1 2 3 -3 -2 -1 1 3 2 -3 -2 -1
-45
-90
-135
-180
Frequency in Hz

8. 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)

9. 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.

10. Channel with 1Hz bandwidth passes first five harmonics
sec -2 -1 1 2 3 4 5 6 7 8
-0.5
0.5
1.5
2.5
volts
sec -2 -1 1 2 3 4 5 6 7 8
-0.5
0.5
1.5
2.5
volts
Channel with
1Hz bandwidth
0.6
0.5
0.4
0.3
0.2
0.1
Frequency in Hz
Magnitude
in volts

11. Channel with 2Hz bandwidth passes first ten harmonics
sec -2 -1 1 2 3 4 5 6 7 8
-0.5
0.5
1.5
2.5
volts
sec -2 -1 1 2 3 4 5 6 7 8
-0.5
0.5
1.5
2.5
volts
Channel with
2Hz bandwidth
0.6
0.5
0.4
0.3
0.2
0.1
Frequency in Hz
Magnitude
in volts

12. Channel with 3Hz bandwidth passes first fifteen harmonics
sec -2 -1 1 2 3 4 5 6 7 8
-0.5
0.5
1.5
2.5
volts
sec -2 -1 1 2 3 4 5 6 7 8
-0.5
0.5
1.5
2.5
volts
Channel with
3Hz bandwidth
0.6
0.5
0.4
0.3
0.2
0.1
Frequency in Hz
Magnitude
in volts