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13.5 Digital Filtering in the Time Domain using none tointegrate none for web,windows applicationcode 39 creating Suppose that you have none for none a signal that you want to lter digitally. For example, perhaps you want to apply high-pass or low-pass ltering to eliminate noise at low or high frequencies respectively; or perhaps the interesting part of your signal lies only in a certain frequency band, so that you need a bandpass lter. Or, if your measurements are contaminated by 60 Hz power-line interference, you may need a notch lter to remove only a narrow band around that frequency.

This section speaks particularly about the case in which you have chosen to do such ltering in the time domain. Before continuing, we hope you will reconsider this choice. Remember how convenient it is to lter in the Fourier domain.

You just take your whole data record, FFT it, multiply the FFT output by a lter function H .f /, and then do an inverse FFT to get back a ltered data set in time domain. Here is some additional background on the Fourier technique that you will want to take into account.

Remember that you must de ne your lter function H .f / for both positive and negative frequencies, and that the magnitude of the frequency extremes is always the Nyquist frequency 1=.2 /, where is the sampling interval.

The magnitude of the smallest nonzero frequencies in the FFT is 1=.N /, where N is the number of (complex) points in the FFT. The positive and negative frequencies to which this lter are applied are arranged in wraparound order.

If the measured data are real, and you want the ltered output also to be real, then your arbitrary lter function should obey H . f / D H .f / .

You can arrange this most easily by picking an H that is real and even in f .. GS1 Bar Codes Glossary 13. Fourier and Spectral Applications If your chosen H .f / none none has sharp vertical edges in it, then the impulse response of your lter (the output arising from a short impulse as input) will have damped ringing at frequencies corresponding to these edges. There is nothing wrong with this, but if you don t like it, then pick a smoother H .

f /. To get a rst-hand look at the impulse response of your lter, just take the inverse FFT of your H .f /.

If you smooth all edges of the lter function over some number k of points, then the impulse response function of your lter will have a span on the order of a fraction 1=k of the whole data record. If your data set is too long to FFT all at once, then break it up into segments of any convenient size, as long as they are much longer than the impulse response function of the lter. Use zero-padding, if necessary.

You should probably remove any trend from the data, by subtracting from it a straight line through the rst and last points (i.e., make the rst and last points equal to zero).

If you are segmenting the data, then you can pick overlapping segments and use only the middle section of each, comfortably distant from edge effects. A digital lter is said to be causal or physically realizable if its output for a particular timestep depends only on inputs at that particular timestep or earlier. It is said to be acausal if its output can depend on both earlier and later inputs.

Filtering in the Fourier domain is, in general, acausal, since the data are processed in a batch, without regard to time ordering. Don t let this bother you! Acausal lters can generally give superior performance (e.g.

, less dispersion of phases, sharper edges, less asymmetric impulse response functions). People use causal lters not because they are better, but because some situations just don t allow access to out-of-time-order data. Time domain lters can, in principle, be either causal or acausal, but they are most often used in applications where physical realizability is a constraint.

For this reason we will restrict ourselves to the causal case in what follows.. If you are still favo ring time-domain ltering after all we have said, it is probably because you have a real-time application for which you must process a continuous data stream and wish to output ltered values at the same rate as you receive raw data. Otherwise, it may be that the quantity of data to be processed is so large that you can afford only a very small number of oating operations on each data point and cannot afford even a modest-sized FFT (with a number of oating operations per data point several times the logarithm of the number of points in the data set or segment)..

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