# 0.6 Digital filtering and the dft

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Digital filtering is not simply converting from analog to digital filters; it is a fundamentally different way of thinkingabout the topic of signal processing, and many of the ideas and limitations of the analog methodhave no counterpart in digital form

—R. W. Hamming, Digital Filters , 3d ed., Prentice Hall 1989

Once the received signal is sampled, the real story of the digital receiver begins.

An analog bandpass filter at the front end of the receiver removes extraneous signals (for instance, it removes televisionfrequency signals from a radio receiver) but some portion of the signal from otherFDM users may remain. While it would be conceptually possible to remove all but the desired user at the start,accurate retunable analog filters are complicated and expensive to implement. Digital filters, on the other hand, are easy to design,inexpensive (once the appropriate DSP hardware is present) and easy to retune.The job of cleaning up out-of-band interferences left over by the analog BPF can be left to the digital portion ofthe receiver.

Of course, there are many other uses for digital filters in the receiver, and this chapter focuses on how to “build” digital filters.The discussion begins by considering the digital impulse response and the related notion of discrete-time convolution.Conceptually, this closely parallels the discussion of linear systems in Chapter [link] . The meaning of the DFT (discrete Fourier transform) closely parallels the meaning ofthe Fourier transform, and several examples encourage fluency in the spectral analysis of discrete data signals. The finalsection on practical filtering shows how to design digital filters with (more or less) any desired frequency responseby using special M atlab commands.

## Discrete time and discrete frequency

The study of discrete-time (digital) signals and systems parallels that of continuous-time (analog) signals and systems.Many digital processes are fundamentally simpler than their analog counterparts, though there are a few subtletiesunique to discrete-time implementations. This section begins with a brief overview and comparison, and then proceedsto discuss the DFT, which is the discrete counterpart of the Fourier transform.

Just as the impulse function $\delta \left(t\right)$ plays a key role in defining signals and systems in continuous time,the discrete pulse

$\delta \left[k\right]=\left\{\begin{array}{cc}1\hfill & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=0\hfill \\ 0\hfill & \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k\ne 0\hfill \end{array}\right)$

can be used to decompose discrete signals and to characterize discrete-time systems. The pulse in discrete time is considerably more straightforward thanthe implicit definition of the continuous-time impulse function in [link] and [link] . Any discrete-time signal can be written as a linear combination of discrete impulses.For instance, if the signal $w\left[k\right]$ is the repeating pattern $\left\{-1,1,2,1,-1,1,2,1,...\right\}$ , it can be written

$\begin{array}{ccc}\hfill w\left[k\right]=& -& \delta \left[k\right]+\delta \left[k-1\right]+2\delta \left[k-2\right]+\delta \left[k-3\right]-\delta \left[k-4\right]\hfill \\ & +& \delta \left[k-5\right]+2\delta \left[k-6\right]+\delta \left[k-7\right]...\hfill \end{array}$

In general, the discrete time signal $w\left[k\right]$ can be written

$w\left[k\right]=\sum _{j=-\infty }^{\infty }w\left[j\right]\phantom{\rule{4pt}{0ex}}\delta \left[k-j\right].$

This is the discrete analog of the sifting property [link] ; simply replace the integral with a sum, and replace $\delta \left(t\right)$ with $\delta \left[k\right]$ .

Like their continuous-time counterparts, discrete-time systems map input signals into output signals.Discrete-time LTI (linear time-invariant) systemsare characterized by an impulse response $h\left[k\right]$ , which is the output of the system when the input is an impulse,though, of course, [link] is used instead of [link] . When an input $x\left[k\right]$ is more complicated than a single pulse, the output $y\left[k\right]$ can be calculated by summing all the responses to all the individual terms, and this leadsdirectly to the definition of discrete-time convolution:

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