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Alias Operator

Aliasing occurs when a signal is undersampled. If the signal sampling rate $ f_s$ 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.

Figure 7.11: Example of frequency-domain aliasing due to undersampling in the time domain.

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 $ N$ -sample signals $ x\in\mathbb{C}^N$ is defined by

\hbox{\sc Alias}_{L,m}(x) &\isdef & \sum_{l=0}^{L-1} x\left(m+lM\right),\\
m &=& 0,1,2,\ldots,M-1,\\

Like the $ \hbox{\sc Downsample}_L()$ operator, the $ \hbox{\sc Alias}_L()$ operator maps a length $ N=LM$ signal down to a length $ M$ signal. A way to think of it is to partition the original $ N$ samples into $ L$ blocks of length $ M$ , with the first block extending from sample 0 to sample $ M-1$ , the second block from $ M$ to $ 2M-1$ , etc. Then just add up the blocks. This process is called aliasing. If the original signal $ x$ 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.


\hbox{\sc Alias}_2([0,1,2,3,4,5]) &=& [0,1,2] + [3,4,5] = [3,5,7] \\
\hbox{\sc Alias}_3([0,1,2,3,4,5]) &=& [0,1] + [2,3] + [4,5] = [6,9]

The alias operator is used to state the Fourier theorem7.4.11)

$\displaystyle \hbox{\sc Downsample}_L \;\longleftrightarrow\;\frac{1}{L}\hbox{\sc Alias}_L.

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

Figure: Illustration of aliasing in the frequency domain. a) $ \hbox{\sc Repeat}_3(X)$ from Fig.7.9c. b) First half of the original unit circle (0 to $ \pi $ ) wrapped around the new, smaller unit circle (which is magnified to the original size). c) Second half ($ \pi $ to $ 2\pi $ ), also wrapped around the new unit circle. d) Overlay of components to be summed. e) Sum of components (the aliased spectrum). f) Both sum and overlay.

Figure 7.12 shows the result of $ \hbox{\sc Alias}_2$ applied to $ \hbox{\sc Repeat}_3(X)$ 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 $ z=1$ (upper right corner of Fig.7.12a). Then the $ \hbox{\sc Alias}_2$ 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 $ \hbox{\sc Alias}_2$ of the original spectrum on the unit circle. To alias by $ 3$ , 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 $ f_s$ , all other unit circles (Fig.7.12b-c) cover frequencies 0 to $ f_s/2$ .

In general, aliasing by the factor $ K$ corresponds to a sampling-rate reduction by the factor $ K$ . 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 $ [-f_s/(2K),f_s/(2K)]$ so that all frequency components which would alias are first removed.

Conceptually, in the frequency domain, the unit circle is reduced by $ \hbox{\sc Alias}_2$ 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 $ f(n)$ is said to be even if $ f(-n)=f(n)$ .

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

Definition: A function $ f(n)$ is said to be odd if $ f(-n)=-f(n)$ .

An odd function is also called antisymmetric.

Note that every finite odd function $ f(n)$ must satisfy $ f(0)=0$ .7.12 Moreover, for any $ x\in\mathbb{C}^N$ with $ N$ even, we also have $ x(N/2)=0$ since $ x(N/2)=-x(-N/2)=-x(-N/2+N)=-x(N/2)$ ; that is, $ N/2$ and $ -N/2$ index the same point when $ N$ is even (since all indexing in $ \mathbb{C}^N$ is modulo $ N$ ).

Theorem: Every function $ f(n)$ can be uniquely decomposed into a sum of its even part $ f_e(n)$ and odd part $ f_o(n)$ , where

f_e(n) &\isdef & \frac{f(n) + f(-n)}{2} \\
f_o(n) &\isdef & \frac{f(n) - f(-n)}{2}.

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

$\displaystyle f_e(n) + f_o(n) = \frac{f(n) + f(-n)}{2} + \frac{f(n) - f(-n)}{2} = f(n).

To show uniqueness, let $ f(n) = f'_e(n) + f'_o(n)$ denote some other even-odd decomposition. Then $ f(n)+f(-n) = 2f_e(n) = 2f'_e(n)
\,\,\Rightarrow\,\,f_e(n)=f'_e(n)$ , and $ f(n)-f(-n) = 2f_o(n) = 2f'_o(n)
\,\,\Rightarrow\,\,f_o(n)=f'_o(n)$ .

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 $ (-)$ :

(+)\cdot(+) &=& (+)\\
(-)\cdot(-) &=& (+)\\
(+)\cdot(-) &=& (-)\\
(-)\cdot(+) &=& (-)

Example: $ \cos(\omega_k n)$ , $ n\in\mathbb{Z}$ , is an even signal since $ \cos(-\theta) = \cos(\theta)$ .

Example: $ \sin(\omega_k n)$ is an odd signal since $ \sin(-\theta) = -\sin(\theta)$ .

Example: $ \cos(\omega_k n)\cdot\sin(\omega_l n)$ is an odd signal (even times odd).

Example: $ \sin(\omega_k n)\cdot\sin(\omega_l n)$ is an even signal (odd times odd).

Theorem: The sum of all the samples of an odd signal $ x_o$ in $ \mathbb{C}^N$ is zero.

Proof: This is readily shown by writing the sum as $ x_o(0) + [x_o(1) + x_o(-1)]
+ \cdots + x(N/2)$ , where the last term only occurs when $ N$ is even. Each term so written is zero for an odd signal $ x_o$ .

Example: For all DFT sinusoidal frequencies $ \omega_k=2\pi k/N$ ,

$\displaystyle \sum_{n=0}^{N-1}\sin(\omega_k n) \cos(\omega_k n) = 0, \; k=0,1,2,\ldots,N-1.

More generally,

$\displaystyle \sum_{n=0}^{N-1}x_e(n) x_o(n) = 0,

for any even signal $ x_e$ and odd signal $ x_o$ in $ \mathbb{C}^N$ . In terms of inner products5.9), we may say that the even part of every real signal is orthogonal to its odd part:

$\displaystyle \left<x_e,x_o\right> = 0

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2016-05-31 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University