Filtros Digitales

FILTER BASICS

Filter Basics
Filters are devices (or algorithms) which change the spectrum of signals - their most prevalent action on signals is to boost, attenuate or completely block frequencies. In the tutorial about sinusoids, we saw that any signal can be seen a sum of sinusoids, each of which having its own frequency f , amplitude A and phase ϕ. In general terms, filters modify theamplitudes and phases of incoming sinusoids according to their frequency. Assume that the input signal to our filter is a single sinusoid with frequency f , amplitude A and phase ϕ. The output signal will again be a sinusoid of frequency f but possibly with different values for the amplitude and phase. And this is already the essence of a filter: all a (linear) filter can do to a sinusoid is to multiplyits amplitude by some factor and to add some offset to its phase; but a filter will never change the general shape of the sinusoid, nor will it change its frequency. So, let’s call that multiplication factor for the amplitude G (for gain) and let’s call that phase shift θ (a greek lowercase theta). Both, gain and phase-shift depend on the frequency of the incoming sinusoid, so both G and θ can beexpressed functions of the frequency f : G = G(f ) and θ = θ(f ) (1)

These two functions are called the magnitude response and the phase response of the filter respectively. Taken together, these two functions form the frequency response of the filter - in a somewhat sloppier slang, one also often hears the term frequency response when the magnitude response alone is meant because this part of thefrequency response often counts more. When we know these two functions, we can can predict the output signal of the filter for any arbitrarily shaped input signal by viewing the input signal as a sum of sinusoids.

Ideal Filters
The classical purpose of a filter is to let certain frequencies pass unchanged and to block others - hence the name filter. An idealized lowpass-filter for example would passall frequencies up to some cutoff-frequency fc and block all frequencies above that cutoff-frequency. The idealized lowpass magnitude response GLP (f ) would be therefore: 1 for f ≤ fc GLP (f ) = (2) 0 for f > fc As we do not see any reason to intentionally introduce phase shift, we would want the phase response of the ideal filter to be identically zero for all frequencies: θ(f ) = 0 (3)

Idealhighpass filters, on the other hand, should pass frequencies above the cutoff frequency and block anything below. Bandpass filters should pass everything within some frequency interval between a lower cutoff frequency fl and an upper cutoff frequency fu . Bandreject filters are the opposite of bandpass filters - they block everything between fl and fh and let everything outside this interval passunchanged. For bandreject filters with a very narrow rejection interval, one also often uses the term ’notch-filter’. The magnitude responses of ideal highpass, bandpass and bandreject filters would be: GHP (f ) = 0 for f < fc 1 for f ≥ fc GBP (f ) = 1 for fl ≤ f ≤ fu 0 otherwise 1 GBR (f ) = 0 for fl < f < fu (4) 1 otherwise

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Real Filters

FILTER BASICS

Wewould like the phase response to be identically zero for the filter types as well in the ideal case. In figure 1 we see the ideal magnitude responses for such filters. The lowpass and highpass filters in these plots are normalized in the sense that they have a cutoff frequency of unity. The physical unit in which frequency is measured is actually irrelevant for this discussion, but if you prefer to dealwith something concrete and practical, feel free to suppose it to be kHz. The bandpass and bandreject filters have a lower cutoff of 0.5 and an upper cutoff of 1.5.

(a) Lowpass

(b) Highpass

(c) Bandpass

(d) Bandreject

Figure 1: magnitude responses of idealized filters

Real Filters
Unfortunately (or maybe not), we do not live in a perfect world and these ideal frequency responses as...