Zero Padding

Zero padding consists of extending a signal (or spectrum) with zeros to extend its time (or frequency band) limits. It maps a length $N$ signal to a length $M>N$ signal, but $M$ need not be an integer multiple of $N$. To unify the time-domain and frequency-domain definitions of zero-padding, it is necessary to regard the original time axis $[0,1,\dots,N-1]$ as indexing positive-time samples from 0 to $N/2-1$ (for $N$ even), and negative times in the interval $n\in[N-1,N-N/2+1]\equiv[-1,-N/2+1]$. Furthermore, we require $x(N/2)\equiv x(-N/2)=0$ when $N$ is even, while odd $N$ requires no such restriction.

Definition:

$$\hbox{\sc ZeroPad}_{M,m}(x) \isdef \left\{\begin{array}{ll} x(m),... ...ert m\vert < N/2 \\ [5pt] 0, & \mbox{otherwise} \\ \end{array} \right. \protect$$ (7.3)

where $m=0,\pm1,\pm2,\dots,\pm M_h$, with $M_h\isdef (M-1)/2$ for $M$ odd, and $M/2 - 1$ for $M$ even. For example,

$$\hbox{\sc ZeroPad}_{10}([1,2,3,4,5]) = [1,2,3,0,0,0,0,0,4,5].$$

In this example, the first sample corresponds to time 0, and five zeros have been inserted between the samples corresponding to times $n=2$ and $n=-2$.

Figure 7.5 illustrates zero padding from length $N=5$ out to length $M=11$. Note that $x$ and $n$ could be replaced by $X$ and $k$ in the figure caption.

Figure: Illustration of zero padding: a) Original signal (or spectrum) $x=[3,2,1,1,2]$ plotted over the domain $n\in [0,N-1]$ where $N=5$ (i.e., as the samples would normally be held in a computer array). b) $\hbox{\sc ZeroPad}_{11}(x)$. c) The same signal $x$ plotted over the domain $n\in [-(N-1)/2, (N-1)/2]$ which is more natural for interpreting negative times (frequencies). d) $\hbox{\sc ZeroPad}_{11}(x)$ plotted over the zero-centered domain.

When a signal $x(n)$ is causal, that is, $x(n)=0$ for all negative-time samples ( $n=-1,-2,\dots,-N/2$), then zero-padding can be carried out by simply appending zeros to the original signal. For example,

$$\hbox{\sc ZeroPad}_{10}([1,2,3,4,5]) = [1,2,3,4,5,0,0,0,0,0]$$   (causal case)$$.$$

The causal definition is natural when $x(n)$ represents a signal starting at time 0 and extending for $N$ samples. On the other hand, when we are zero-padding a spectrum, or we have a time-domain signal including both positive- and negative-time samples, then the zeros should be inserted ``outside the nonzero interval'' of the signal or spectrum, as defined in Eq. (7.3).