The DFT shifting theorem
Contents
6.2. The DFT shifting theorem#
We’ve seen that for signals
But what can we say about phase for more general signals? In particular, if we have an arbitrary signal
Before we continue down this line of thought, we must first establish what it means to delay a signal under our periodicity assumption. This leads us to the notion of a circular shift.
Definition 6.1 (Circular shifting)
Let
to be the circular shift of
When it is clear from context that shifting is circular, we may drop the “
Circular shifting can seem like a strange thing to do: taking samples from the end of the signal and putting them at the beginning?
However, it is a natural consequence of combining our periodicity assumption with the definition of delay.
Indeed, it can lead to some strange behaviors if
Fig. 6.2 A repeating signal
Given this definition of shifting (delay), we can say the following about its effect on the DFT spectrum of a signal.
Theorem 6.1 (DFT shifting)
Let
The DFT spectrum of
This says that no matter what delay we use, it’s always possible to exactly predict the spectrum of the delayed signal from the spectrum of the input signal.
The proof of the shifting theorem is purely algebraic, and relies on the periodicity assumption of
Proof. If we have an index
The last step follows because even though the summation ranges from
6.2.1. What does this do?#
It’s worth taking some time to understand the shifting theorem. Not only does it say exactly how the spectrum changes, as a signal shifts in time, but the change itself is also interesting.
Note that the argument of the exponential is a purely imaginary number:
This means that multiplication implements a rotation of
Of course, this is exactly what we would hope should happen: delay only changes the horizontal (time) position of a signal, not its amplitude.
A slightly more subtle point is that each DFT component
Example 6.1
We can visualize the shifting theorem by examining what happens to each DFT component
Fig. 6.3 Top: a test signal
Fig. 6.3 illustrates several key features of circular shifting.
First, observe that the different DFT components rotate at different speeds when the delay
Note also that the motion of each
6.2.2. What does this not do?#
The shifting theorem implies that if a signal is circularly shifted, its magnitude spectrum is unchanged:
However, the converse is not true: two signals with the same magnitude spectrum may not be related by a circular shift of one another.
Example 6.2
Let
so the two signals have the same magnitude spectrum.
However, there is no delay
Fig. 6.4 Left: an impulse (top), a negative impulse (middle), and a 3-step delay (bottom).
Center: the DFT spectrum (real and imaginary) components of each signal are distinct.
Right: all three signals have identical magnitude spectra
Tip
Phase is important. Having the same magnitude spectrum is not enough to ensure that two signals are the same except for a phase shift.