Steve Smith (@SteveSmith)
Benford's law solved with DSP
I have a longtime interest in the mystery of 1/f noise. A few years ago I came across Benford’s law, another puzzle that seemed to have many of the same characteristics.
Suppose you collect a large group of seemingly random numbers, such as might appear in a newspaper or financial report. Benford’s law relates to the leading digit of each number, such as "4" in 4.268, "3" in 0.0312, and "9" in -932.34. Since there are nine possible leading digits...
Waveforms that are their own Fourier Transform
Mea Culpa
There are many scary things about writing a technical book. Can I make the concepts clear? It is worth the effort? Will it sell? But all of these pale compared to the biggest fear: What if I'm just plain wrong? Not being able to help someone is one thing, but leading them astray is far worse.
My book on DSP has now been published for almost ten years. I've found lots of typos, a few misstatements, and many places where the explanations confuse even me. But I have been lucky;...
An Interesting Fourier Transform - 1/f Noise
Power law functions are common in science and engineering. A surprising property is that the Fourier transform of a power law is also a power law. But this is only the start- there are many interesting features that soon become apparent. This may even be the key to solving an 80-year mystery in physics.
It starts with the following Fourier transform:
The general form is tα ↔ ω-(α+1), where α is a constant. For example, t2 ↔...
I misspoke... Taking the mean of each 1/3 octave should make the white noise flat. Taking the sum of each 1/3 octave should make the 1/f noise flat. In other words,...
As a first approximation, they will be the same, in spite of the different number of cycles in the dataset. However, there are a bunch of second order effects...
You aren't calculating the energy correctly. You want to square each FFT value (to convert from amplitude to energy), and then take the average of the points...
Here's some information on the parabolic phase... last topic in the chapterhttp://www.dspguide.com/CH11.PDF
I think this is the correct place and you will find lots of people that can help. The problem is that your question isn't expressed very well... it isn't clear...
Hi Alon,Here's some material that may help... Chapters 14-21 deal with the practical aspects of filter design. Since you asked about recursive filters, you...
Hi Nelson,Audio is a 1D signal... a series of numbers. Your approach is not correct... you can't convert a 1D signal into a 2D signal by using the bit patterns...
This is an example where DSP can really shine... using software to overcome limitations of hardware. Unfortunately, the solution to this particular problem is not...
Here's a tidbit that will strike home...I took two years of graduate level DSP courses in the 1980's from Tom Stockham. He seldom wrote an equation on the board......
No, it's not possible in general. In an arbitray second-order filter there is an interaction between the parameters that cannot be duplicated in a simple cascade...
Hi Max,Say you make the gain on frame one different from frame two, which produces a jump between them. You probably will want a way to smooth-out this discontinuity. ...
Interference pattern... cool! Since you are doing so well, now explain this hexagon at the north pole of Saturn
Nice... Hendrix was decades ahead of his time. Here's a couple of links on some filters you might like to try. http://www.dspguide.com/CH19.PDFhttp://www.dspguide.com/CH16.PDFOn...
Hi Dan,I've never used this technique, so let me see if I understand the approach... I might be all wrong. The idea is to use a low frequency signal to adjust some...
Looks much better!Convolution is a single well-defined mathematical operation, taking two signals and creating a third. However, there are a number of different...
Hi Jag,You have the general understanding of the problem, but there are several math and conceptual errors. You start off with the frequency response of a continuous...
"Can I just multiply my coefficients by 4 to reach the same output amplitude?"Yes. In fact, that is how I design most of my FIR filters... Design the shape, and...
Music is usually 20Hz-20kHz, but voice is only 200 Hz to 3.2kHz. My guess is that your software doesn't handle as low of frequency signals as you suspect. The...
An audio signal has zero DC, meaning that it goes both positive and negative with respect to ground. However, electronic devices often have only +5v available....
Linear interpolation is probably fine for this. Loop through each of the 2048 values you want to calculate in the converted data, say, x = 1 to 2048. Then for...
Can I assume if I'm using a natural / standard IR, (ex. of some physical space), that there shouldn't be any problems during the convolution / deconvolution of the...
Hi Nelson,I don't think you misled me; I think you have the same problem I've tried to describeI assume that the following signal is the convolution of your dry...
Hey Rick!Yes, it bothers me too, but it took me awhile to realize why... probably like everyone else. That in itself says it's not a good problem for a basic geometry...
I've given it some more thought. Here's a possible way for you to obtain an impulse response that is reasonably random, but has no zeros in its frequency response. ...
ok, I see what you are trying to do. You want to security encrypt an audio signal by convolving it with white noise, and then unencrypt it by deconvolution. ...
1. Say you have some time domain signal, such as from a scientific instrument, which contains some random noise. A common problem is that the signal has been unavoidably...
You wrote: "I could've sworn my Professor thought that we could deconvolve a signal along the time domain (undoing the time shifts and amplitude changes caused...
You can do that if you are directly designing the transfer function, such as in filter design. But that doesn't have much practical application. Usually deconvolution...
The problem is that most transfer functions have frequencies where the value is zero, preventing the input signal from being calculated. Even if the transfer function...
Hey Rick,Yes, that's correct. Keeping this large electric field manageable is one of the key challenges as IC feature sizes have gone down, resulting in the...
I like demos to prove the point... maybe some sort of instrumentation, audio or image processing.Say, simulate a soldier speaking into a microphone on a battlefield,...
Are you trying to create a filter with an arbitrary phase response that you can select? Or are you trying to theoretically duplicate the phase response given by...
This may help. Fig. 17-2 shows what frequency response you can achieve with FIR filters of different lengths. http://www.dspguide.com/CH17.PDF
Here's a couple of links you may find useful... my experience with this very issue. http://www.dspguide.com/CH28.PDFhttp://www.dspguide.com/CH29.PDF
That helps, I see what you are trying to do. I'm nearly certain that your problem is a bug in the code, not an accuracy issue in determining the phase. You are...
Keeping things simple, making a few assumptions, and building on your idea of using the FFT--A problem you will have with the FFT is that the frequency of your sine...
So let's say I move the anti-aliasing hardware filter to a cutoff of 300Hz. Then I generate a set of digital band-pass filter coefficients with a sharp roll-off...
Hi Nelson,Glad to help; everyone starts out at some point. Your last post is mixing up the concepts of time domain, frequency spectra, and spectrogram. I think you...
Hi NelsonYes, all correct. The adversary could look at the frequency spectra of the dry and wet signals, note that there was a gap in the information, and try...
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