Doing DFT without using FFT function (2024)

The problem with the code that you suggested was that it requires initialization of data array and/or a finite number of test data beforehand.

For my application I intended test data to be transmitted via the serial port every few millisecond which means that the array used in storage of incoming ECG data cannot be initialized beforehand. And that is why I had to use the variable 'a' to double as a loop counter.

If I had used different variables as counters, the computation of DFT will fail since 'N' in your code would have a fixed value when in fact it varies from 1 to 500 with time.

The storage array, X is actually an array of complex numbers depending on the computation of the DFT, which is why I used abs(X) to capture the magnitude in the variable 'mag' and used to display out on the graph.

Am I missing something from my understanding? Thanks for the swift reply as well!

You do need a finite and definite number of data values to carry out a DFT (or FFT). You can set N to as many data values as you have at any point in time, and carry out the operation on those data, but the number of values can't be indefinite in any sense.

You set mag to abs(X(a,1)). This is just a scalar - what happens to all the other values you've computed for X, from X(1) to X(a-1)? In any case, I can't see where you use the value of mag.

I've now noticed that fftECG is actually given a value, using fft() - I'd missed that earlier.

The bottom line is that any version of the DFT (either the fft function or the code with loops) operates on a vector with a definite number of data points. If the data comes in a value at a time, you need to simply store these in an array until you have enough of them to do the analysis you want, then carry out the DFT on that array.

I would agree on what you have said mathematically but I used the built in fft function on one incoming value from the serial port and it managed to compute fft in real time, because the figures are able to be updated live.

This is where I used the fft function after collecting data from the serial port 'arduino':

and also

Output onto a graph is done here:

But however when I try to represent this built in function with a self-written DFT code as per discussed earlier, the calculated values in the array X turns out totally different from the values calculated by the inbuilt fft() function.

Which is the reason why I questioned if the equation I've input is wrong.. Or is there is something else that the inbuilt fft() does besides calculating DFT?

No, the inbuilt fft() does exactly what the looping DFT code does, except that it is much faster and there may be tiny numerical differences in the result.

The difference is due to the fact that you are making use of different variables when you use the inline DFT and you call fft(). For example, fft() takes the number of data to be the length of ecg, but you set the number of data to be a.

Note that there is simply no way to compute a DFT without a vector of data to apply it to, and that vector will have a definite length. As I said, the DFT code you used is, in itself, correct.

By the way, what is the length of ecg?

Hm, it is strange because ecg is just a double. It isn't even an array so there isn't a length to speak of. Which hence prompted me to believe fft() does not actually apply to a fixed vector.

On a side note, I had to use a in each of the loops because in computation of DFT, a is synonymous to the sample size N, which you had pointed out correctly should be defined beforehand conventionally. However, because the size of dataFft is always increasing as the serial port receives an additional data point, N (and therefore a) will always be increasing as well if I were to carry out real-time data collection instead of declaring the length beforehand.

I am puzzled because the fft() function was able to work this out on its own without me declaring a fixed length for it.

This was the code BEFORE I implemented the DFT hard-coded formula and worked perfectly fine:

And because fft() is a shortcut function and has no academic value whatsoever, a hardcoded formula is preferred for teaching purposes, despite the fact that fft() will be computationally faster than any form of hardcoded DFTs.

ah, I found out where it went wrong. ecg should be stored as an array and not be taken as a singular value.

Many thanks!

That sounds right. fft() will work on a single value (a scalar) - it just treats it as a vector of length 1, and returns its argument. This is correct mathematically, since the DFT of a scalar is just the scalar itself.