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.
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 ,
for any even signal and odd signal in . In terms of inner products (§5.9), we may say that the even part of every real signal is orthogonal to its odd part: