How to decompose a signal into its contributory signals

Ok, so in the interest of full disclosure at the outset, I'm a 69 y.o. retired epidemiologist who has an interest in HRV signal analysis specifically, and physics as a whole--always have. Am not an EE, physicist or mathematician so be kind, if you will. This is my first post.

In signal analysis (let's assume EM signal), I do have (I believe) a basic understanding of the use of Fourier analysis to decompose the signal/function into its oscillatory components via sine and cosine functions. And, if desired, to use Fourier synthesis to reverse the process to obtain the original signal. Where I am "lost" is that these "component" sine and cosine signals obviously don't represent the actual contributory signals which sum to produce the signal being examined. E.g., we have a given acoustic EM signal consisting of various overlaying frequencies and amplitudes.

Think a musical composition. How does one "dissect" the actual contributing component signal frequencies and amplitudes (I realize there are other qualities such as tonality and timbre in music which I'm not concerned about here) that are resulting in the final observed/listened-to signal. Seemingly, just because one can decompose a signal into sine/cosine functions does not indicate that those functions actually exist within the studied signal.

I hope I am getting my question across and apologize to those readers who are more well-versed in this topic than I am. I will leave this be before even professing more than I presume to understand at this point. Thank you kindly.

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anorlunda
Staff Emeritus

Seemingly, just because one can decompose a signal into sine/cosine functions does not indicate that those functions actually exist within the studied signal.

I disagree. Yes they do. That is like saying that when adding integers that 2 does not actually contain 1 and 1.

Mathematical models depict reality. Some with more accuracy. Some with extended range. But to say that they aren't real because they are mathematical is incorrect.

Hello Triquetra - Welcome -

I believe you are on the correct track - a Fourier Analysis ( Transform) can be used to convert a signal from time domain ( in a way how we experience it) - to frequency domain. This is a mathematical conversion of data - regardless of the source(s).

But there really is no way to take a mixed signal an break it back up into the unknown original inputs. When some info is known about the original signals - there are some clever tricks that may kind of do this - but each case of signal and original known info needs to be evaluated as their own special case.

Music is a good example of how complex of a problem this can be. There are programs today that can listen to a musical track, identify the vocal track - and then remove that from the signal. But I do not think this is not real time filter or process - but analysis of the recording and then re-processing it. And - I am sure- is far from perfect, causing distortion and leaving artifacts of the vocal content in the output..

So HRV - is a good field for this processing. For example constructing a "model" pulse(cycle) for a subject ( since you can general get a lot of samples - you have some statistical models to narrow or widen), and then removing that "model" from every pulse. There are of course many standard abnormalities that are easily detected, and once identified, can be eliminated from the "signal" similarly as the voice in audio. This is a very big field of study - and Apple, Fit Bit - and those guys are right there.

jrmichler
Mentor
Fourier transforms include one huge assumption - that the signal you are analyzing is periodic on the sample interval. You are analyzing a signal of finite duration. The Fourier analysis assumes that the "real" signal is an infinite series of your finite signal placed end to end. For a good example, calculate the FFT of a pure sine wave. First do it for an integer number of cycles, 2 or 3 cycles is enough. Then add a half or quarter cycle and repeat the FFT. The frequency has not changed, but the FFT is completely different. It is different because one signal is continuous and smooth, while the other has discontinuities.

Fourier transforms can be used to study music by doing FFT's of short pieces of the composition. The result is called a spectrogram. A spectrogram is the result of a tradeoff between FFT's of many short pieces to get the changes, and longer pieces to get better frequency resolution.

Does HRV stand for heart rate variability? If so, Fourier transforms may not be the best analysis tool. I'm not sure exactly what would be the best tool for that. I would first look into wavelet transforms. After that, I would try creating a short waveform representing one heartbeat, then doing a traveling cross correlation down the length of your measured signal and recording the time of each peak in the cross correlation.

essenmein
jim hardy
Gold Member
Dearly Missed
Seemingly, just because one can decompose a signal into sine/cosine functions does not indicate that those functions actually exist within the studied signal.

My freshman year EE professor showed us boys a graphical method to derive the first few Fourier terms from an arbitrary waveform.
He gave each of us a sheet pf graph paper with a plot of an arbitrary waveform ,
had us derive the first five Fourier terms
then plot those terms on graph paper with same scale ..
Indeed the result strongly resembled the original in shape and it was apparent that if we continued adding more terms the fidelity would get better.

That made me a believer in Fourier.

Some months later i lived for a while with the frame and strings from a junked piano in the room.
Its strings would audibly vibrate in sympathy when ambient noise got the least bit loud.
and it dawned on me that its strings amounted to an array of sharply tuned filters
each responding to whatever component of background noise happened to be at that string's frequency-
a "Poor Man's Fourier Frequency Analyzer" if you will.

So i do believe at the gut level those frequencies ARE indeed present in signals.
And math agrees.

old jim

Glad someone mentioned the assumed "periodicity" of a signal for a FFT to be valid. To get around taking FFT of non periodic signals (ie basically all signals since they are all at some point finite), "window" functions are used:

https://en.wikipedia.org/wiki/Window_function

jrmichler and jim hardy
Baluncore
Welcome to PF.

Analysing signals requires an understanding of the cause and resonance mechanism of signals. A musical note is a sinewave, usually with integer harmonics.

Musical notes will be exponentially distributed in frequency, 12 per octave, with tuning errors. They will not be periodic within your sample but window functions will fix that problem. A note will have a time envelope. For a piano or drum that will be an impulse followed by an exponentially decaying wave. For the human voice it will be an envelope containing a possibly varying frequency. Both will have physical structure dependent harmonics.

When a hammer strikes a resonator, a muscle contracts, or a valve closes to stop a flow, a resonance may result. You must look for the finger print of that resonance. To do that you need to perform correlation.

Correlation is done by converting a reference sample of a known individual signal and the combined signal to the frequency domain by using an FFT. Then multiply frequency by frequency before converting the product back to the time domain with the inverse FFT. You will see spikes at different times that show when the reference pattern occurred in the combined signal. As you identify and find individual signals you can subtract them from the record. Continue until only noise remains.

Similar tricks such as autocorrelation will identify echos in a signal.

jrmichler
Think a musical composition. How does one "dissect" the actual contributing component signal frequencies and amplitudes (I realize there are other qualities such as tonality and timbre in music which I'm not concerned about here) that are resulting in the final observed/listened-to signal. Seemingly, just because one can decompose a signal into sine/cosine functions does not indicate that those functions actually exist within the studied signal.
The main problem for those, who come across with FFT for the first time is that our senses are just always tells otherwise. Music is a perfect example for this. What our ears does is actually something closely resembling to FFT, but what we hear is something what is based on a complex, always learning, context-sensitive neural network making associations. Thus, we will hear music (with all the details) instead of a raw FFT data feed.
There is no way such strong override from sensory experience to not cause confusion. It helps a lot if you keep in mind that even when you hear music - your ears working on a real-time, quite sophisticated and tricky biological FFT the same time.

jrmichler and anorlunda
sophiecentaur
Gold Member
2020 Award
So i do believe at the gut level those frequencies ARE indeed present in signals.
More than that. Neither of the two domains is more 'correct' than the other; they are mutually equivalent. It all depends on the context as to which domain we choose to use to describe a signal. If we look at a musical signal from a CD player with an oscilloscope, we see (and believe in) a time varying voltage. If we take an HF Comms receiver and tune it up and down the band, we see a set of peaks in received signal strength (the stations) and we believe that we are seeing a set of frequencies. In both cases, we could analyse what we are studying with 'the other' system. For instance a musician would hear that musical (time domain) signal in terms of notes, chords and keys, all of which are frequency related descriptions; the scope trace would mean very little to her. Turn the gain of a wide band HF pre-amp up and we would see a noise-like variation of (micro)volts on the antenna feed.
What makes it all more difficult is the fact that all real signals have limited time and frequency windows so what we are 'really' aware of is a combination of frequency and time information.

jim hardy and jrmichler
marcusl
Gold Member
There is much more to the validity of Fourier analysis than has been expressed so far in this thread. It is possible to show rigorously that sines and cosines are the fundamental characteristic functions for any linear time-invariant (LTI) system. That is, so long as the system (guitar string, flute, audio amplifier, electronic filter, etc.) is linear (not overdriven into distortion) and time-invariant (no one is stretching the string tighter with the tuning knob while playing, or changing the air density while blowing through the flute, or modifying the amplifier gain or filter circuit dynamically during a measurement), then all signals passing through or emanating from the system may be described by a sum of sines and cosines. Physicists usually call these by the German-derived word "eigenfunctions," but I think that the obsolete English name for them (characteristic functions) is more descriptive.

This is also why a frequency response curve characterizes so much of an LTI system. By sweeping the frequency of an input signal and measuring the amplitude and phase response of the output, you are literally performing a complete Fourier analysis.

berkeman
sophiecentaur
Gold Member
2020 Award
There is much more to the validity of Fourier analysis than has been expressed so far in this thread.
I have already made the point that time and frequency domain descriptions of signals are equally valid. I don't know what more validity you could ask for. I would say that you, yourself, are implying that the 'direction of' the Fourier Transform has some basic significance. There is none and it works both ways. There is no correct domain in which to describe a signal.

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marcusl
Gold Member
I was trying to state, in a non-mathematical way, why Fourier analysis with its sines and cosines is so appropriate for systems we commonly deal with. It is a valid question, since other systems require other basis sets for proper analysis (Hermite polynomials for quantum vibrations within a square potential well, for example).

berkeman
sophiecentaur
Gold Member
2020 Award
I was trying to state, in a non-mathematical way, why Fourier analysis with its sines and cosines
That's a bit of an oxymoron. Sines and Cosines are very mathematical functions so you're in Maths up to your neck from the start. It's a common phenomenon that we tend to be very selective about what things we accept as 'understandable' and what things are "too hard guv'nor". The sinusoidal graphs that people happily plot in order to produce sure waves and triangular waves can lull us into a false sense of understanding.
There are a host of other valid ways of analysing a time dependent function into harmonic sets of other functions.

berkeman
Mentor