Convolution and Time Domain
Filtering
Cuthbert Nyack
The above diagram shows the equivalence between the time and
frequency domains. Passing a signal through a filter is equivalent to
taking the convolution of the input signal with the impulse
response of the filter.
Fn = 1 in the applet below shows a noisy sine being convolved with
the impulse response of a filter. Since the sine has a single
frequency, then a band pass filter centered at the frequency
of the sine is used. The bandwidth of the filter can be modified
by W but its effect is limited because the impulse response is windowed
with a raised cosine to reduce 'edge' effects in the convolution.
Fn = 2 to 5 show special cases of Fn = 1.
Fn = 6 is the case with 2 sinusoids. Here the filter and impulse
response used can be set by 'C'. C = 1 uses 2 bandpass filters
centered at w and w1.
This filter can be used to reduce the noise while passing both
frequencies. To pass only w, C = 0 and
C = 2 to pass w1.
Fn = 7 to 13 show special cases of Fn = 6.
Fn = 14 shows the case of an AM signal with modulating index 'm'
carrier frequency w and modulating
frequency w1. A bandpass filter is used.
Fn = 15 to 18 show special cases of Fn = 14.
For Fn = 17 a narrow BW filter is used which reduces the sidebands.
Fn = 19 shows the case of a DSB signal with
carrier frequency w and modulating
frequency w1.
Fn = 20 to 21 show special cases of Fn = 19.
For Fn = 21 a narrow BW filter is used which reduces the sidebands.
Fn = 22 shows the case of an AM signal with modulating index 'm'
carrier frequency w and modulating
frequency w1.
A double bandpass filter is used which can pass the 2 sidebands and reduce
the carrier.
Fn = 23 to 24 show special cases of Fn = 22. Both cases show the
AM being converted to a DSB signal.
Fn = 25 shows the case of an PM signal with modulating index 'm1'
carrier frequency w and modulating
frequency w1. A bandpass filter is used.
Fn = 26 to 28 show special cases of Fn = 25. Fn = 27 uses a
narrow bandpass filter which only passes the carrier.
Fn = 29 shows the case of a square wave and a low pass filter.
Fn = 30 to 34 show special cases of Fn = 29.
Fn = 35 shows the case of a square wave and a band pass filter.
Fn = 36 to 43 show special cases of Fn = 35 corresponding to
nh = 1, 2, 3, 4, 5, 6, 7 and 9. nh sets the center frequency
of the bandpass filter at the
harmonic nh w.
In this case convolution is used as a spectrum analyser to
find the spectral components of the square wave.
Fn = 44 shows the case of a rectangular wave and a band pass filter.
Fn = 45 to 55 show special cases of Fn = 44 corresponding to
nh = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. nh sets the center frequency
of the bandpass filter at the
harmonic nh w.
In this case convolution is used as a spectrum analyser to
find the spectral components of the rectangular wave.
Fn = 56 shows the case of a triangle wave and a low pass filter.
Fn = 57 to 58 show special cases of Fn = 56.
Fn = 59 shows the case of 2 triangle waves with fundamental
frequencies w and w1
and with overlapping spectra. Individual waves are in
pale yellow and their resultant in red.
To separate the 2 a comb filter
with a |cos| frequency response is used. The impulse response of this
filter is shown in orange. In this case the the comb is set to
pass components at nw ie the higher
frequency triangle wave.
Fn = 60 shows the case where the comb is set to
pass components at nw1 ie the lower
frequency triangle wave.
Fn = 61, 62 and 63, 64 show the same effect with slightly
different number of zeros for the comb filter.
Fn = 65 shows the frequency response of the |cos| comb filter
used above and a comb filter consisting of a series of rectangles.
The latter should be less sensitive to noise. However when the
impulse response is windowed the frequency response becomes broader
than that shown.
Fn = 66 to 72 show special cases of the latter comb filter.
Image below shows the noise in the green signal removed by
convolving it with the impulse response of a double band pass filter
in orange.
Cyan curve shows the result of the convolution. Convolving
the clean red signal with the orange impulse response results
in the pink signal.
The possibility of separating 2 frequency overlapping triangle
waves by convolution with the impulse response of a comb
filter is illustrated by the image below.
Red signal shows the sum of the 2 triangles. When it is
convolved with the orange impulse response, the result is the
pink triangle wave. When the noisy version of the red
signal shown in green is convolved with the orange signal, the
result is shown in cyan.
Note that the magnitude of the impulse response of the
comb filter is the same
as the optical pattern obtained from a multislit aperture.
Changing the response of the comb filter results in the other
triangle as shown below.
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COPYRIGHT © 1997, 2012 Cuthnbert A. Nyack.