B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

42 SIGNALS AND SIGNAL SPACE
and
For the series in Example 2.3, for instance,
Do = 0.504
D1 = -:: = 0.l22e-j75·96 0 ==} ID1 I = 0.122, LD1 = -75.96 °
0.504 '75.960 0
D-1 = --. = 0.122e' ==} ID-1 I = 0.122, LD_1 = 75.96
1-14
- 0.504 - -j82.87 ° -
D2 - l + j8
- 0.0625e ==} ID2 I - 0.0625, LD2 = -82.87 °
0.504 82.87 ° 0
D-2 = --. = 0.0625e' ==} ID-2 1 = 0.0625, LD_2 = 82.87
1 - 18
and so on. Note that D n and D-n are conjugates, as expected [see Eq. (2.63b)].
Figure 2.14 shows the frequency spectra ( amplitude and angle) of the exponential Fourier
series for the periodic signal <p(t) in Fig. 2.13b.
We notice some interesting features of these spectra. First, the spectra exist for positive
as well as negative values of J (the frequency). Second, the amplitude spectrum is an even
function of J and the angle spectrum is an odd function of J. Equations (2.63) show the
symmetric characteristics of the amplitude and phase of D n .
What Does Negative Frequency Mean?
The existence of the spectrum at negative frequencies is somewhat disturbing to some people
because by definition, the frequency (number of repetitions per second) is a positive quantity.
How do we interpret a negative frequency Jo? We can use a trigonometric identity to express
a sinusoid of a negative frequency -Jo by borrowing wo = 2nJo, as
cos (-wot + 0) = cos (wot - 0)
Figure 2.14
Exponential
Fourier spectra
for the signal in
Fig. 2.13a.
0.504
0.122 (a)
0.0625
-10
-8
-6 -4 -2 2 4 6 8 10
0)-
L Dn
···· .........
7t
2
(b)
-10
8 -6 -4 -2 2
-
4 6 8 10
..l .l...... J ...... 1 l
0)-
-7t
2