Digital Signal Processing Reference
In-Depth Information
are harmonic, say what the fundamental frequency is.
x
1
(t) = 2 cos(27t) + 3 cos(235t)
x
2
(t) = 2 cos(2
2
t) + 3 cos(2t)
x
3
(t) = 2 cos(27t + ) + 3 cos(2t=4)
Answer:
To be harmonically related, each sinusoid must have a frequency that is an
integer multiple of some fundamental frequency.
That is, they should be in the
following form. Remember that k is an integer.
N
X
x(t) =
a
k
cos(2kf
0
t +
k
)
k=1
So we put the above signals in this form, and see if the k values are integers,
which they must be for the sinusoids to be harmonically related. Looking at the
arguments of the cosine function for x
1
, let k
1
and k
2
be values of k.
27t = 2k
1
f
0
t +
k
, Let
k
= 0
f
0
= 7=k
1
235t = 2k
2
f
0
t +
k
, Let
k
= 0
f
0
= 35=k
2
Substituting in f
0
, we get
35=k
2
= 7=k
1
k
1
=k
2
= 7=35
k
1
=k
2
= 1=5
So if k
1
= 1, then k
2
= 5
f
0
= 35=k
2
= 35/5 = 7 Hz.
Therefore, x
1
(t) has integer k values, and is harmonic. x
3
(t) is, too. x
2
(t)
does not have integer k values (due to the frequency of ) and is, therefore, not
harmonic. The third signal, x
3
, is a bit tricky because the frequency components
are not lined up as we may expect.f
0
= 1 Hz, a
0
= 1, a
k
=f3; 0; 0; 0; 0; 0; 2g, and
k
=f=4; 0; 0; 0; 0; 0;g.
