Doug Kerr
Well-known member
This is a rerun (hopefully refined) of a discussion I gave a while ago in response to a question.
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We hear of image-processing programs that use deconvolution to, in effect, "back out of" the image captured on the sensor the effects of such unwanted behavior of the optical system as defocus.
Deconvolution is, not surprisingly, the inverse of the mathematical operation of convolution. First, I'll explain convolution as it applies not to optical "signals" but rather to electrical ones.
The time and frequency domains
We can describe electrical "signals", and what happens to them, in either the time domain or the frequency domain, which are said to be duals of each other. We can get a simple glimpse of this by considering a situation in which a light flashes every 0.5 seconds. Its period is 0.5 seconds (a time domain description), and its frequency is 2.0 times per second (2.0 hertz) (a frequency domain description).
The time domain description of an electrical "signal" is its waveform, a plot of its instantaneous voltage vs. time.
The frequency domain description of the signal is its spectrum. One form of this is its power density spectrum (the "power spectrum"). This tells us how the power in the signal is distributed among components at different frequencies. It is plotted as a function of frequency.
The power density spectrum is accompanied, to give the full description of the signal, by a plot of phase vs. frequency. This tells us the relative time relationship of the individual frequency components. From her on, I will mostly ignore the phase aspects of things, not that it isn't important, but again to help the basic concept to be more concisely discussed.
The amplitude spectrum
When we work mathematically with the spectrum of a signal, it turns out that we must usually work with a plot whose ordinate is the square root of the power density spectrum curve, called the amplitude spectrum.
Transforming between domains
If we know the waveform (time domain description) of a signal, we can apply the mathematical operation called the Fourier transform to that and the result will be the amplitude (and phase) spectrums of the signal (the frequency domain description).
If we know the amplitude and phase spectrum of a signal (its frequency domain description) we can apply the mathematical operation called the inverse Fourier transform to that and get the waveform (time domain description) of the signal.
Filters
Anything the signal, passes through that changes its characteristics can be considered to be a filter. We will limit ourselves to linear filters, which we can think of as filters whose relative effect on the signal is not changed by the actual amplitude ("largeness") of the signal. If we are interested in what happens to the signal generated by a studio microphone as it passes through the mixing console, we can treat the console as a filter.
The nature of a filter
In the frequency domain
For our purposes here, a filter can be described by two functions of frequency (plots vs frequency):
• The amplitude frequency response. This tells us the ratio of the output of the filter (in amplitude - that is voltage - terms) to the input, plotted vs. frequency (that is, it tells us what that ratio would be for a signal that contains only a certain frequency.
• The phase frequency response. This tells us, for any frequency, how a signal component at that frequency will have its relative time shifted by passing through the filter. Yes, we will ignore this from now on.
In the time domain
We can also completely describe the response of a filter by imagining that we introduce into it a "unit impulse", a single electrical pulse of zero length in time but containing, over its (zero-length) life one unit of energy (obviously a mathematical fiction). We then note the waveform of the output signal (that is, a plot of its instantaneous voltage vs. time).
This is called the impulse response of the filter. If we know that, we know everything we need to predict what will happen to any signal passing through the filter.
Predicting the output of the filter
Working in the frequency domain
We have a signal for which we know the amplitude spectrum. We pass it through a filter (that could be an amplifier which we treat as a filter for this purpose) whose amplitude response is known. What will the output signal be like?
We multiply the amplitude spectrum of the original; signal by the amplitude response of the filter. By this we main that at every frequency in the range of interest, we multiply the value of the amplitude spectrum by the value of the amplitude response. The result will be a value for every frequency in the range. The plot of that will be the amplitude spectrum of the output signal (its frequency domain description). (For a signal with a "continuous" spectrum, won't there be an infinite number of multiplications? Yes.)
[Continued]
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We hear of image-processing programs that use deconvolution to, in effect, "back out of" the image captured on the sensor the effects of such unwanted behavior of the optical system as defocus.
Deconvolution is, not surprisingly, the inverse of the mathematical operation of convolution. First, I'll explain convolution as it applies not to optical "signals" but rather to electrical ones.
The time and frequency domains
We can describe electrical "signals", and what happens to them, in either the time domain or the frequency domain, which are said to be duals of each other. We can get a simple glimpse of this by considering a situation in which a light flashes every 0.5 seconds. Its period is 0.5 seconds (a time domain description), and its frequency is 2.0 times per second (2.0 hertz) (a frequency domain description).
The time domain description of an electrical "signal" is its waveform, a plot of its instantaneous voltage vs. time.
The frequency domain description of the signal is its spectrum. One form of this is its power density spectrum (the "power spectrum"). This tells us how the power in the signal is distributed among components at different frequencies. It is plotted as a function of frequency.
Those familiar with the concept of density functions will recognize that my description is a little simplistic. I will often depart from rigor here in the interest of illuminating the concepts.
The power density spectrum is accompanied, to give the full description of the signal, by a plot of phase vs. frequency. This tells us the relative time relationship of the individual frequency components. From her on, I will mostly ignore the phase aspects of things, not that it isn't important, but again to help the basic concept to be more concisely discussed.
The amplitude spectrum
When we work mathematically with the spectrum of a signal, it turns out that we must usually work with a plot whose ordinate is the square root of the power density spectrum curve, called the amplitude spectrum.
Transforming between domains
If we know the waveform (time domain description) of a signal, we can apply the mathematical operation called the Fourier transform to that and the result will be the amplitude (and phase) spectrums of the signal (the frequency domain description).
If we know the amplitude and phase spectrum of a signal (its frequency domain description) we can apply the mathematical operation called the inverse Fourier transform to that and get the waveform (time domain description) of the signal.
Filters
Anything the signal, passes through that changes its characteristics can be considered to be a filter. We will limit ourselves to linear filters, which we can think of as filters whose relative effect on the signal is not changed by the actual amplitude ("largeness") of the signal. If we are interested in what happens to the signal generated by a studio microphone as it passes through the mixing console, we can treat the console as a filter.
The nature of a filter
In the frequency domain
For our purposes here, a filter can be described by two functions of frequency (plots vs frequency):
• The amplitude frequency response. This tells us the ratio of the output of the filter (in amplitude - that is voltage - terms) to the input, plotted vs. frequency (that is, it tells us what that ratio would be for a signal that contains only a certain frequency.
• The phase frequency response. This tells us, for any frequency, how a signal component at that frequency will have its relative time shifted by passing through the filter. Yes, we will ignore this from now on.
In the time domain
We can also completely describe the response of a filter by imagining that we introduce into it a "unit impulse", a single electrical pulse of zero length in time but containing, over its (zero-length) life one unit of energy (obviously a mathematical fiction). We then note the waveform of the output signal (that is, a plot of its instantaneous voltage vs. time).
This is called the impulse response of the filter. If we know that, we know everything we need to predict what will happen to any signal passing through the filter.
Predicting the output of the filter
Working in the frequency domain
We have a signal for which we know the amplitude spectrum. We pass it through a filter (that could be an amplifier which we treat as a filter for this purpose) whose amplitude response is known. What will the output signal be like?
We multiply the amplitude spectrum of the original; signal by the amplitude response of the filter. By this we main that at every frequency in the range of interest, we multiply the value of the amplitude spectrum by the value of the amplitude response. The result will be a value for every frequency in the range. The plot of that will be the amplitude spectrum of the output signal (its frequency domain description). (For a signal with a "continuous" spectrum, won't there be an infinite number of multiplications? Yes.)
[Continued]