Digital Signal Processing Reference
In-Depth Information
introduction; this topic is covered extensively in Chapter 7, \The Number e." For
example, to nd and plot the magnitude spectrum of x(t) = 2 + 2 cos(2(200)t), we
put it in terms of cosine functions.
x(t)
=
2 + 2 cos(2200t)
=
2 cos(0) + 2 cos(2200t)
Next, use inverse Euler's formula:
cos() = (e
j
+ e
j
)=2:
Notice how this results in two frequency components: one at the frequency given,
and one at the negative frequency. For the spectrum, this implies that the range of
frequencies will be dierent from what we saw with the previous example. Actually,
the information stays the same, only our spectral plot has dierent, but equivalent,
frequency values. This is explained in Chapter 6, \The Fourier Transform."
2(e
j0
+ e
j0
)=2 + 2(e
j2(200)t
+ e
j2(200)t
)=2
x(t)
=
2(1 + 1)=2 + (2=2)(e
j2(200)t
+ e
j2(200)t
)
=
2 + e
j2(200)t
+ e
j2(200)t
=
This results in a magnitude of 2 at frequency 0 Hz, and a magnitude of 1 at
frequencies 200 Hz and -200 Hz. This is shown in the magnitude plot, Figure 4.14.
2
1
1
−300
−200
−100
0
100
200
300
Hz
Figure 4.14: Spectrum plot: magnitude of x(t) = 2 + 2 cos(2(200)t).
Example:
For the signal x(t) below,
x(t) = 4 cos(2100t + 3=4) + 2 cos(2200t) + 5 cos(2300t3=5)
draw the magnitude spectrum and the phase spectrum of this signal.
