Signal manipulations

Signal manipulations are generally decomposed of a few basic transformations. These transformations may be classified either as those that are transformations of the independent variable $n$ or those that are transformations of the amplitude of $x\left[n\right]$, i.e. the dependent variable.

The most common types of amplitude transformations are addition, multiplication and scaling: $$\begin{array}{ccc}\text{Addition}& :& y\left[n\right]=x_1\left[n\right]+x_2\left[n\right]\\ \text{Multiplication}& :& y\left[n\right]=x_1\left[n\right]\cdot x_2\left[n\right]\\ \text{Scaling}& :& y\left[n\right]=c\cdot x\left[n\right]\end{array}$$ Performing these operations is straightforward and involves only point-wise operations on the signal.

Sequences are often altered and manipulated by modifying the independent function variable $n$, where $f\left[n\right]$ is some function of the index $n$. If, for some value of $n$, $f\left[n\right]$ is not an integer, $y\left[n\right]=x\left[f\right[n\left]\right]$ is undefined. The most common transformations include shifting (delaying or advancing), reversal, and time-scaling as summarized in the following equations: $$\begin{array}{ccc}\text{Time shifting (delay or advance)}& :& f\left[n\right]=n-n_0\\ \text{Time reversal}& :& f\left[n\right]=-n\\ \text{Time scaling = Down- or Up- sampling}& :& f\left[n\right]=M\cdot n\text{or}f\left[n\right]=n/M\end{array}$$ Determining the effect of modifying the index $n$ may always be accomplished using a simple tabular approach of listing, for each value of the index $n$, the value of $f\left[n\right]$ and then setting $y\left[n\right]=x\left[f\right[n\left]\right]$. However, for many transformations this is not necessary, and the sequence may be determined or plotted directly. Examples of delaying, time-reversal and time scaling, such as down sampling and up-sampling are illustrated in Fig. 1.

Note that shifting, reversal, and time scaling operations are order dependent. Therefore, one needs to be careful in evaluating compositions of these operations. Fig. 2 shows an example in which a signal $x\left[n\right]$ is first delayed over $n_0$ samples, resulting in $x[n-n_0]$. Reversing this signal, that is replacing the running index $n$ by $-n$, results in the signal $x[-n-n_0]$. Switching the order of these two operations, that is first reverse and then delay results in another signal namely $x[-n+n_0]$.

Finally the delta pulse may be used to decompose an arbitrary signal $x\left[n\right]$ into a, possible infinite, sum of weighted and shifted delta pulses: $$x\left[n\right]=\sum _{k=-\infty }^{\infty }x\left[k\right]\delta [n-k]$$ Each term in the sum, $x\left[k\right]\delta [n-k]$, is a signal that has amplitude $x\left[k\right]$ at index $n=k$ and a value of zero for all other values of the index $n$.