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68 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
2. Time shifting: A shift in the time domain corresponds to the phase
shifting.
F [x(n − k)] = X(e jω )e −jωk (3.6)
3. Frequency shifting: Multiplication by a complex exponential corresponds
to a shift in the frequency domain.
F [ x(n)e jω0n] = X(e j(ω−ω0) ) (3.7)
4. Conjugation: Conjugation in the time domain corresponds to the
folding and conjugation in the frequency domain.
F [x ∗ (n)] = X ∗ (e −jω ) (3.8)
5. Folding: Folding in the time domain corresponds to the folding in the
frequency domain.
F [x(−n)] = X(e −jω ) (3.9)
6. Symmetries in real sequences: We have already studied the conjugate
symmetry of real sequences. These real sequences can be decomposed
into their even and odd parts, as discussed in Chapter 2.
x(n) =x e (n)+x o (n)
Then
F [x e (n)] = Re [ X(e jω ) ]
F [x o (n)] = j Im [ X(e jω ) ] (3.10)
Implication: If the sequence x(n) isreal and even, then X(e jω )is
also real and even. Hence only one plot over [0,π]isnecessary for its
complete representation.
A similar property for complex-valued sequences is explored in
Problem P3.7.
7. Convolution: This is one of the most useful properties that makes
system analysis convenient in the frequency domain.
F [x 1 (n) ∗ x 2 (n)] = F [x 1 (n)] F [x 2 (n)] = X 1 (e jω )X 2 (e jω ) (3.11)
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