Alias Operator

*Aliasing* occurs when a signal is *undersampled*. If the signal
sampling rate
is too low, we get *frequency-domain
aliasing*.

The topic of aliasing normally arises in the context of
*sampling* a continuous-time signal. The *sampling theorem*
(Appendix D) says that we will have no aliasing due to sampling
as long as the sampling rate is higher than twice the highest
frequency present in the signal being sampled.

In this chapter, we are considering only discrete-time signals, in order to keep the math as simple as possible. Aliasing in this context occurs when a discrete-time signal is downsampled to reduce its sampling rate. You can think of continuous-time sampling as the limiting case for which the starting sampling rate is infinity.

An example of aliasing is shown in Fig.7.11. In the figure,
the high-frequency sinusoid is indistinguishable from the
lower-frequency sinusoid due to aliasing. We say the higher frequency
*aliases* to the lower frequency.

Undersampling in the frequency domain gives rise to *time-domain
aliasing*. If time or frequency is not specified, the term
``aliasing'' normally means frequency-domain aliasing (due to
undersampling in the time domain).

The *aliasing operator* for
-sample signals
is defined by

Like the
operator, the
operator maps a
length
signal down to a length
signal. A way to think of
it is to partition the original
samples into
blocks of length
, with the first block extending from sample 0
to sample
,
the second block from
to
, etc. Then just add up the blocks.
This process is called *aliasing*. If the original signal
is
a time signal, it is called *time-domain aliasing*; if it is a
spectrum, we call it *frequency-domain aliasing*, or just
aliasing. Note that aliasing is *not invertible* in general.
Once the blocks are added together, it is usually not possible to
recover the original blocks.

**Example:**

The alias operator is used to state the Fourier theorem (§7.4.11)

That is, when you downsample a signal by the factor , its spectrum is aliased by the factor .

Figure 7.12 shows the result of
applied to
from Figure 7.9c. Imagine the spectrum of
Fig.7.12a as being plotted on a piece of paper rolled
to form a cylinder, with the edges of the paper meeting at
(upper
right corner of Fig.7.12a). Then the
operation can be
simulated by rerolling the cylinder of paper to cut its circumference in
half. That is, reroll it so that at every point, *two* sheets of paper
are in contact at all points on the new, narrower cylinder. Now, simply
add the values on the two overlapping sheets together, and you have the
of the original spectrum on the unit circle. To alias by
,
we would shrink the cylinder further until the paper edges again line up,
giving three layers of paper in the cylinder, and so on.

Figure 7.12b shows what is plotted on the first circular wrap of the cylinder of paper, and Fig.7.12c shows what is on the second wrap. These are overlaid in Fig.7.12d and added together in Fig.7.12e. Finally, Figure 7.12f shows both the addition and the overlay of the two components. We say that the second component (Fig.7.12c) ``aliases'' to new frequency components, while the first component (Fig.7.12b) is considered to be at its original frequencies. If the unit circle of Fig.7.12a covers frequencies 0 to , all other unit circles (Fig.7.12b-c) cover frequencies 0 to .

In general, aliasing by the factor
corresponds to a
*sampling-rate reduction* by the factor
. To prevent aliasing
when reducing the sampling rate, an *anti-aliasing lowpass
filter* is generally used. The lowpass filter attenuates all signal
components at frequencies outside the interval
so that all frequency components which would alias are first removed.

Conceptually, in the frequency domain, the unit circle is reduced by
to a unit circle
*half* the original size, where the two halves are summed. The inverse
of aliasing is then ``repeating'' which should be understood as
*increasing* the unit circle circumference using ``periodic
extension'' to generate ``more spectrum'' for the larger unit circle.
In the time domain, on the other hand, downsampling is the inverse of
the stretch operator. We may interchange ``time'' and ``frequency''
and repeat these remarks. All of these relationships are precise only
for *integer* stretch/downsampling/aliasing/repeat factors; in
continuous time and frequency, the restriction to integer factors is
removed, and we obtain the (simpler) *scaling theorem* (proved
in §C.2).

Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.

**Definition: **A function
is said to be *even* if
.

An even function is also *symmetric*, but the
term symmetric applies also to functions symmetric about a point other
than 0
.

**Definition: **A function
is said to be *odd* if
.

An odd function is also called *antisymmetric*.

Note that every finite odd function
must satisfy
.^{7.12} Moreover, for any
with
even, we also have
since
; that is,
and
index
the same point when
is even (since all indexing in
is modulo
).

**Theorem: **Every function
can be
uniquely
decomposed into a sum of its even part
and odd part
, where

*Proof: *In the above definitions,
is even and
is odd by construction.
Summing, we have

To show uniqueness, let denote some other even-odd decomposition. Then , and .

**Theorem: **The product of even functions is even, the product of odd functions
is even, and the product of an even times an odd function is odd.

*Proof: *Readily shown.

Since even times even is even, odd times odd is even, and even times odd is odd, we can think of even as and odd as :

**Example: **
,
, is an
*even* signal since
.

**Example: **
is an *odd* signal since
.

**Example: **
is an *odd* signal (even times odd).

**Example: **
is an *even* signal (odd times odd).

**Theorem: **The sum of all the samples of an odd signal
in
is zero.

*Proof: *This is readily shown by writing the sum as
, where the last term only occurs when
is even. Each
term so written is zero for an odd signal
.

**Example: **For all DFT sinusoidal frequencies
,

More generally,

for

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University