Filters. Signals are sequences of numbers. Simple algebraic operations on signals can perform useful functions: shifting multiplication addition

Transcription

1 Filters Signals are sequences of numbers. Simple algebraic operations on signals can perform useful functions: shifting multiplication addition

2 Simple Example... Smooth points to better reveal trend X = [ ]

3 Smoothing Average each pair of adjacent points INPUT OUTPUT X = [ ] Y =[ ] plot (X,'o-') plot (Y,'o-') for k = 2:length(X) Y(k) = (X(k) + X(k-1)) / 2; end for k = 2:length(X) Y(k) =.5*X(k) +.5*X(k-1); end

4 Image Filtering Images are matrices, values specifies intensity (or RGB) of pixel Average adjacent intensities in rows and in columns Original Average adjacent pixels

5 Original Average adjacent pixels

6 Filtering Audio: buzz Original After two-point averaging 200 Hz pulse train 200 Hz pulse train Time in ms Time in ms

7 Pulse Train can be generated as a sum of phasors that are integer multiples of f0, each with equal amplitude.

8 Generating a pulse train Phasor function [signal, t] = make_buzz_phasor(srate, f0) t = 0:1/srate:.5; signal = zeros (1, length(t)); for freq = f0 : f0: srate/2 omega = 2*pi*freq; signal = signal + cos(omega*t); end Time function approach function [buzz, t] = make_buzz(srate, f0) dur =.5; totsamps = dur*srate; samps_per_cycle = fix((1/f0) * srate); buzz = (rem([0:totsamps], samps_per_cycle) == 0); t = (1:length (buzz))/srate;

9 Alternative Conceptualization of shift weight (multiply) add Averaging.5 * X = [ ].5 * X! = [ ].5*X = [ ].5*X!= [ ] Y = [ ]

10 Generalized Feedforward Filter (H) multiple of shifted versions of input (M) weighting coefficients associated with each (b) b is a vector of coefficients of length M, weighting the M shifted versions of the input Y(k) = b(1)* X(k) + b(2)* X(k-1) b(m)* X(k-(M-1)) In the averaging example, b=[.5.5] As M increases, signal gets smooth (if b>0)

11 Differencing Suppose b = [.5 -.5] This takes the difference between adjacent samples (which is the velocity) Opposite effect from smoothing--removes overall trend while emphasizing the noise plot (Y, 'o-') plot (X,'o-')

12 Differencing Images Original Difference

13 Differencing the buzz... Original After two-point difference 200 Hz pulse train 200 Hz pulse train Time in ms Time in ms

14 Frequency Response What happens if the input to an averaging filter is a phasor (sinusoid)? We know that it will be a sinusoid of the same frequency and amplitude as the input, but different amplitude and phase. Why? Filters delay, multiply and add. So the filter operation will sum the original sinusoid with others of the same frequency and different phases and amplitudes. We have already seen that the sum of sinusoids of the same frequency is itself a sinusoid, whose amplitude and phase can be determined from the amplitude and phase of the sinusoids being summed.

15 Frequency response of two point smoother Y (k) =.5X(k)+.5X(k 1) X(k) =e j k Y (k) =.5e j k +.5e j (k 1) Y (k) =.5e j k +.5e j k j Y (k) =.5e j k +.5e j k e j Means shift input by ω Y (k) =.5(1 + e j ) e j k Y (k) =.5(1 + e jw ) X(k) Y (k) X(k) =.5(1 + e j )=H( ) Transfer Function

16 Transfer Function H is complex number, function of ω: magnitude angle H( ) Arg(H( )) multiply the input phasor by H to get output phasor Y (k) =H( )X(k) abs(h) tells us the amplitude ratio output/input angle(h) tells us the phase shift of the output. Y (k) =H( ) X(k)e jarg(h( )) IF X(k) =e j k Y (k) =H( ) e j( k+arg(h( ))

17 To see what happens to the amplitude of a phasor of arbitrary frequency, we can plot the value of H(ω) for a set of ω values we choose. π radians/sample (=sampling rate/2) is the highest frequency that can be coded in a digital signal. Anything higher will be equivalent to a low frequency (its alias). Amplitude Response radians sample cycles 2 radians sr samples second

18 Phase Response To see what happens to the phase of a phasor of arbitrary frequency, we can plot the value of Arg(H(ω)) for a set of ω values we choose.

19 Code for plotting amplitude, phase response % AVERAGE2PTS_RESP % Louis Goldstein % 31 March 1996 % Use MATLAB to sketch magnitude and phase of H(w) for the simple average2pts filter. % % Set up a vector of test frequencies to plot, from % zero to the Nyquist frequency % Frequencies are represented in radians per sample nfreqs = 100; w = 0:pi/nfreqs:pi-(pi/nfreqs); % %Compute a vector of H values, one for each value of w. H =.5 * (1 + exp(-j*w)); % % Plot the Amplitude of H as a function of frequency: figure (1) plot (w/pi,abs(h),'or') xlabel ('Frequency in fractions of pi, the Nyquist frequency') ylabel ('Amplitude') % % Plot the phase of H as a function of frequency figure(2) plot (w/pi,angle(h)*180/pi, 'or') xlabel ('Frequency in fractions of pi, the Nyquist frequency') ylabel ('Phase shift in degrees')

20 Distributivity of Filters Filters are distributive. So H is a linear filter (feedforward, but also feedback) If: X H Y W H Z Then: X+W H Y+Z

21 Filtering more complex signals Any periodic signal can be decomposed into a sum of phasors (X(ω)), each of which has its amplitude and phase (Fourier Theorem). The amplitude spectrum of the input shows the amplitudes of all the phasors that compose it. We can determine the filter output of a complex signal by summing the filter outputs of all the phasors that compose it. The resulting amplitudes will be shown in the amplitude spectrum of the output.

22 H(ω) Buzz input spectrum (f0 = 200 Hz) output spectrum