Shift Theorem

Theorem: For any

and any integer

,
Proof:
The shift theorem is often expressed in shorthand as
The shift theorem says that a
delay in the time domain corresponds to
a
linear phase term in the
frequency domain. More specifically, a
delay of

samples in the time waveform corresponds to the
linear
phase term

multiplying the
spectrum, where

.
7.13Note that spectral magnitude
is unaffected by a linear phase term. That is,

.
Linear Phase Terms
The reason

is called a
linear phase term is
that its phase is a linear function of frequency:
Thus, the
slope of the phase, viewed as a linear function of
radian-frequency

, is

. In general, the
time
delay in samples equals
minus the slope of the linear phase
term. If we express the original
spectrum in polar form as
where

and

are the magnitude and phase of

, respectively
(both real), we can see that a linear phase term only modifies the
spectral
phase 
:
where

. A positive time delay (waveform shift to
the right) adds a
negatively sloped linear phase to the original
spectral phase. A negative time delay (waveform shift to the left) adds a
positively sloped linear phase to the original spectral phase. If we
seem to be belaboring this relationship, it is because it is one of the
most useful in practice.
In practice, a
signal may be said to be
linear phase
when its phase is of the form
where

is any real constant (usually an integer), and

is an
indicator function which takes on the
values 0 or

over the points

,

.
An important class of examples is when the signal is regarded as a
filter impulse response.
7.14 What all such
signals have in common is that they are
symmetric about the time

in the time domain
(as we will show on the next page). Thus, the term ``linear phase
signal'' often really means ``a signal whose phase is linear between

discontinuities.''
A
zero-phase signal is thus a
linear-phase signal for which the
phase-slope

is zero. As mentioned above (in
§
7.4.3), it would be more precise to say ``0-or-

-phase
signal'' instead of ``zero-phase signal''. Another better term is
``zero-centered signal'', since every real (even)
spectrum corresponds
to an even (real) signal. Of course, a zero-centered symmetric signal
is simply an
even signal, by definition. Thus, a ``zero-phase
signal'' is more precisely termed an ``even signal''.
In practical
spectrum analysis, we most often use the
Fast
Fourier Transform7.15 (FFT) together with a
window function

. As discussed
further in Chapter
8, windows are normally positive (

),
symmetric about their midpoint, and look pretty much like a ``
bell
curve.'' A window multiplies the
signal 
being analyzed to form a
windowed signal

, or

, which
is then analyzed using an FFT. The window serves to
taper the
data segment gracefully to zero, thus eliminating spectral
distortions
due to suddenly cutting off the signal in time. Windowing is thus
appropriate when

is a short section of a longer signal (not a
period or whole number of periods from a
periodic signal).
Theorem: Real symmetric FFT windows are
linear phase.
Proof: Let

denote the window samples for

.
Since the window is symmetric, we have

for all

.
When

is odd, there is a sample at the midpoint at time

. The midpoint can be translated to the time origin to
create an even signal. As established on page
![[*]](../icons/crossref.png)
,
the
DFT of a real and even signal is real and even. By the shift
theorem, the DFT of the original symmetric window is a real, even
spectrum multiplied by a
linear phase term, yielding a
spectrum
having a phase that is linear in frequency with possible
discontinuities of

radians. Thus, all odd-length real
symmetric signals are ``linear phase'', including FFT windows.
When

is even, the window midpoint at time

lands
half-way between samples, so we cannot simply translate the window to
zero-centered form. However, we can still factor the window
spectrum

into the product of a linear phase term
![$ \exp[-\omega_k(M-1)/2]$](http://www.dsprelated.com/josimages_new/mdft/img1391.png)
and a real spectrum (verify this as an
exercise), which satisfies the definition of a linear phase signal.
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