The Cooper Union
Department of Electrical Engineering
ECE111
Signal Processing & Systems Analysis
Problem Set I: Signals & Spectra
SOLUTION KEY
Spring 2010
1.
Figure 1
shows a
sawtooth
waveform with period
T
and peak amplitude
A
. Compute
its RMS value. Does the answer depend on
T
?
RMS value=
h
1
T
R
T
0
x
2
(
t
)
dt
i
1
=
2
.
We have
x
(
t
) =
A
T
t
. So:
k
x
k
RMS
=
°
1
T
Z
T
0
A
2
T
2
t
2
dt
±
1
=
2
=
°
A
2
T
3
1
3
T
3
±
1
=
2
=
A
p
3
The answer does not depend on
T
, and is proportional to peak value
A
.
Note the
constant of proportionality is NOT
1
p
2
;
many people mistakenly associate the
1
p
2
factor with the de°nition of RMS value; that only applies for sinusoidal waveforms; as
this example illustrates,
1
p
2
does not yield the RMS scaling factor for all waveforms!
2. Periodicity in discretetime is a bit more complicated than periodicity in continuous
time. Speci°cally, a discretetime signal
x
[
n
]
is considered periodic only if
x
[
n
+
N
] =
x
[
n
]
for some
INTEGER
N
; if the formula "appears±periodic for some noninteger
value
T
, then the signal is not really periodic. In particular, that means discretetime
sinewaves may not be periodic.
Another complication with periodicity is if (say in
continuoustime)
x
has period
T
, then it also has period
2
T
,
3
T
, even
°
T
and so forth
(i.e., if
x
(
t
+
T
) =
x
(
t
)
, then
x
(
t
+
mT
) =
x
(
t
)
for all integers
m
); thus, let us agree
that the period of a signal (whether continuous or discrete) is the
smallest positive
value for which periodicity holds.
(a) In continuoustime, what is the period of
e
jt
?
In discretetime, is
e
jn
periodic,
and, if so, what is the period?
Keep in mind that
exp (
j°
) = 1
IF AND ONLY IF
°
= 2
±n
for integer
n
; period
of
exp (
jt
)
is
T
= 2
±
; on the other hand
exp (
jn
)
is not periodic in general if the
continuous time period is not rational the corresponding discrete time signal is
not periodic!
(b) In continuoustime, what is the period of
cos (3
±t=
4)
? In discretetime, is
cos (3
±n=
4)
periodic and, if so, what is the period?
T
= 2
±=
(3
±=
4) = 8
=
3
; in the discrete time case, the period has to be an integer;
think of the discrete period
N
necessarily being an integer multiple of
T
: this
suggests
N
= 8
.
In other words the discrete time signal repeats after three cycles
of the underlying continuoustime waveform.
Anyway to summarize: continuous
time signal has period
T
= 8
=
3 sec
, discretetime signal has period
N
= 8
samples.
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 Spring '08
 Faculty
 Signal Processing, Cooper Union Department of Electrical Engineering

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