Transcript Digital Signal Processing

The complex exponential
(or two-sided) Fourier series
General information and Example 2.5
Two-Sided or Complex Exponential Form
of the Fourier Series
x (t ) 

j 2 nf o t
c
e
n
n 
1
where cn 
T
to  T
 j 2 nf o t
x
(
t
)
e
dt

to
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)
Two-Sided or Complex Exponential Form
(continued)
x (t ) 

j 2 nf o t
c
e
n
n 
1
where cn 
T
to  T
 j 2 nf o t
x
(
t
)
e
dt

to
Disadvantages:
 Form is not intuitive due to
 Use of complex exponentials
 Complex cn coefficients
 Resultant “negative frequency” components
(continued)
Plotting cn in the Complex Plane
x (t ) 

j 2 nf o t
c
e
n
n 
1
where cn 
T
to  T
 j 2 nf o t
x
(
t
)
e
dt

to
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).
Plotting cn in the Complex Plane
We can express cn in
terms of a real and
imaginary component,
or in terms of a
magnitude |cn| and a
phase fn.
Imaginary
axis
Im{cn}
cn
fn
Re{cn}
Real axis
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
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)
3
2


1
-10
-5
0
5
10
seconds
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
Magnitude
in volts
0.5
0.4
180
Phase in
degrees
0.3
135
0.2
90
0.1
45
1
-3
-2
-1
1
Frequency in Hz
2
3
3
2
1
-45
-90
-135
-180
Frequency in Hz
2
3
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
volts
2.5
volts
2.5
-2
2
2
1.5
1.5
1
1
0.5
0.5
-1
-0.5
1
2
3
4
5
6
7
8
sec
-2
-1
-0.5
1
2
3
4
Channel with
1Hz bandwidth
0.6
0.5
Magnitude
in volts
0.4
0.3
0.2
0.1
Frequency in Hz
0.6
0.5
Magnitude
in volts
0.4
0.3
0.2
0.1
Frequency in Hz
5
6
7
8
sec
Channel with 2Hz bandwidth passes
first ten harmonics
volts
2.5
volts
2.5
-2
2
2
1.5
1.5
1
1
0.5
0.5
-1
-0.5
1
2
3
4
5
6
7
8
sec
-2
-1
-0.5
1
2
3
4
Channel with
2Hz bandwidth
0.6
0.5
Magnitude
in volts
0.4
0.3
0.2
0.1
Frequency in Hz
0.6
0.5
Magnitude
in volts
0.4
0.3
0.2
0.1
Frequency in Hz
5
6
7
8
sec
Channel with 3Hz bandwidth passes
first fifteen harmonics
volts
2.5
-2
volts
2.5
2
2
1.5
1.5
1
1
0.5
0.5
-1
-0.5
1
2
3
4
5
6
7
8
sec
-2
-1
-0.5
1
2
3
Channel with
3Hz bandwidth
0.6
0.5
Magnitude
in volts
0.4
0.3
0.2
0.1
Frequency in Hz
0.6
0.5
Magnitude
in volts
0.4
0.3
0.2
0.1
Frequency in Hz
4
5
6
7
8
sec