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1.Signals
à 1.3 Analog and Digital Frequencies
A complex exponential signal, whether in continuous time or in discrete time, is completely determined
by its amplitude A , phase a , and frequency which can be F0 Hz (or W0 rad/sec) in continuous time or
w0 radians in discrete time, as in the following expressions:
x HtL = Aeja ejW0 t = Aeja ej2pF0 t
x@nD = Aeja ejw0 n
Since, as we anticipated several times, the complex exponential is the actual building block of most
signal analysis tools (the famous reference frame we talked about at the beginning of the section) it is
convenient to represent the complex exponential in terms of its three parameters (Amplitude, Phase and
Frequency). In particular let us represent graphically the complex exponential as two plots (Amplitude
and Phase), as functions of frequency (F0 or W0 in the continuous time case, or w0 in the discrete time
case) as shown in figure 1.3.1.
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A
Magnitude
F0
Phase
F
α
F0
F
Figure 1.3.1: Frequency Domain Representation of a Complex Exponential Signal
When the signal is the superposition (ie sum) of two or more complex exponentials of different amplitudes and phases, we just add them in the plot and place them appropriately in frequency. So for example
a sinusoidal signal, as
A e-ja e-j2pF0 t
x HtL = Acos H2 pF0 t + aL = €€€2A eja ej2pF0 t + €€€
2
is represented by two complex exponentials with frequencies F0 and -F0 respectively, as shown in
figure 1.3.2. Notice the appearance of the negative frequency -F0 which corresponds to the complex
exponential with the same amplitude and opposite phase.
Magnitude
A
A
2
2
− F0
F0
F ( Hz )
Phase
− F0
−α
α
F0
F (Hz )
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Figure 1.3.2: Frequency Domain Representation of a Sinusoidal
Signal
Similarly for the discrete time case. A discrete time sinusoid
x@nD = Acos Hw0 n + aL = €€€A2 eja ejw0 n + €€€A2 e-ja e-jw0 n
is represented by the two plots of magnitude and phase as in figure 1.3.3 . Again it is represented by two
frequencies w0 and -w0 with same amplitude and opposite phase.
Magnitude
A
A
2
2
ω0
− ω0
ω (rad )
Phase
− ω0
−α
α
ω0
ω (rad )
Figure 1.3.3: Frequency Domain Representation of a Sinusoidal Sequence
In both continuous time and discrete time domains, we can start calling this graphical representation
with its own name: the frequency domain representation of the signal. Although at this point we can
understand it only for signals explicitly made of complex exponentials and sinusoids, we can see that it
represents the contribution of each frequency to the total signal.
So, given a complex exponential, or a sinusoidal signal, whether x HtL in continuous time or x@nD in
discrete time, we have seen that we can determine its own frequency domain representation.
Homework Problems: Problems 2.1, 2.2
See if you are following. Click HERE for a Question (not graded)
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Now the question: is there a one-to-one correspondence between each complex exponential signal
x HtL or x@nD and the frequency representation? This is important because, if this is true, then there is
no ambiguity in going from the time domain to the frequency domain. But if this is not true, we have to be
aware of possible ambiguities.
Another way of posing this question is whether we can find two sinusoids having different frequencies
and yet being identical in time.
In continuous time the answer is YES: there is a one to one correspondence between complex exponentials in time and their frequency representation. If we take any two complex exponentials
x1 HtL = ej2pF1 t and x2 HtL = ej2pF2 t with different frequencies F1 ¹ F2 , in the time domain they
are two different signals.
In discrete time, it is not so. Two sequences x1@nD = ejw1 n and x2@nD = ejw2 n with w1 = w2 + k2p ,
and k being ANY integer, are indistinguishable in the time domain. In fact, when this is the case, we can
write
x2 @nD = ejw2 n = ej Hw1 +k2pL n = Hejk2pnL ejw1 n = x1@nD
since both k and n are integers, and therefore ejk2pn = 1.
Similar arguments apply for sinusoidal sequences. If we take two sequences
x1@nD = Acos Hw1 n + a1 L and x2 @nD = Acos Hw2 n + a2L and frequencies and phases are
related by one of the two relationships
w2 = w1 + k2p ,
a2 = a1
w2 = -w1 + k2p ,
a2 = -a1
with k integer, then the two sinusoidal sequences have the same samples. The first case follows from the
same argument we had for the complex exponentials. The second case holds since
x2@nD = Acos H-w1 n + k2p - a1L = Acos Hw1 n - k2p + a1L = x1@nD, for all n
As illustrated in figure 1.3.4. sinusoids with frequencies as shown correspond to the same time domain
sequence x@nD.
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− ω 1 + 2π
−ω
ω
1
ω 1+ 2π
1
ω
0
(rad )
x[n ]
n
Figure 1.3.4: Ambiguity between Discrete Time and Frequency Domain
This shows that we do not need all the digital frequencies -¥ < w < +¥ to represent a discrete time
signal. In fact if we limit the digital frequency w within an interval of length 2 p , say for example
-p < w £ +p , then there is a one to one correspondence between discrete time complex exponential
signals and their frequency representation. Rather than going through a tedious demonstration, the reader
can easily verify that, for each frequency w in the interval H-p, +pD the corresponding "aliases" (ie
the frequencies which yield the same discrete time sequence) are ALL OUTSIDE the interval
H-p, +pD itself.
Example: consider a sinusoid with frequency w0 = 0.2 p . Then the aliases are at frequencies
w0 + k2p = H0.2 + 2 kL p =. .., -1.8 p, 2.2 p, ...
-w0 + k2p = H-0.2 + 2 kL p =. .., -2.2 p, 1.8 p, ...
where we have shown explicitely the alias frequencies for k = -1 and k = +1. As we can see all
aliased frequencies are either smaller than -p or larger that +p , since -p < w0 < +p.
−π
ω 1 ≠ω
2
+π
ω
x1[ n]
≠
(rad )
x2[ n]
Figure 1.3.5: One to One Correspondence of Discrete Time and Frequency Domains
when -p < w £ +p
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This is summarized in the following:
Continuous Time Signals: -¥ < F < +¥
¬ ®
ej2pFt
Discrete Time Signals:
-p £ w < +p
¬ ® ejwn
Homework Problems: Problems 2.3, 2.4, 2.5
See if you are following. Click HERE for a Question (not graded)