### Shift Theorem

**Theorem: **For any
and any integer ,

*Proof: *

The shift theorem is often expressed in shorthand as

*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 .

^{7.13}Note 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:

*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

*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.

#### Linear Phase Signals

In practice, a *signal* may be said to be *linear phase*
when its phase is of the form

*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.''

#### Zero Phase Signals

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''.

#### Application of the Shift Theorem to FFT Windows

In practical spectrum analysis, we most often use the *Fast
Fourier Transform*^{7.15} (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 ,
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 and a real spectrum (verify this as an exercise), which satisfies the definition of a linear phase signal.

**Next Section:**

Convolution Theorem

**Previous Section:**

Symmetry