python plt.specgram the number of xaxis' value when I change nfft, overlap

Question

I am trying to use plt.specgram

But I found something weird in the x-axis setting.

My data is as below..

Date       Value
2018-01-01  0.1
2018-01-02  0.4
...
2020-02-27  0.7
2020-02-28  0.3

The length of data is 821. (The dates I wrote are temporary values)

When I plot the spectrogram as below:

Pxx, freqs, bins, im=ax.specgram(df, NFFT=256, Fs=1.0, noverlap=220, cmap='inferno', mode='psd')
mesh=ax.pcolormesh(bins, freqs, Pxx, norm=LogNorm(), cmap='inferno')

the x-axis only shows values below 700.

1. When I set noverlap = 75, the x-axis shows values over 700.

2. When I set nfft = 256, the x-axis shows values up to about 220.

3. When I set nfft = 256*3, the x-axis becomes up to about 256*2.

If I want to the x-axis to show values up to 821, how do I set the parameters of specgram?
(I want to set nfft = 256, and noverlap = 220)

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This is the fig1

This is the fig2

This is the fig3

Answer 1

Short answer

You can't.

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EDIT

• To answer your question as I understand it, which is

  how can I get the x-axis to span the full length of my data ?

  As @Hilmar mentioned, one way to achieve this is to have windows of length 1, i.e nfft = 1 and overlap = 0 (note, in Matlab, the spectrogram function won't let you input windows of length less than 2). With nfft = 256, the further down the x-axis you'll get is by setting overlap = 68. To understand why that is, and how a spectrogram is displayed, in particular the x-axis, you can go on to the "Long answer" below.

• A slightly different and in my opinion more interesting question is:

  With a fixed window length and overlap, how do I make sure every sample in my data has been windowed and transformed ?

  The answer to that is to make sure that the length $N$ of the input data satisfies $$\left(\frac{N-r}{L-r}\right) = K$$

  $K\in1,2,3,\dots$

  To ensure this condition, choose

  $$K = \bigg\lceil \frac{N_o-r}{L-r} \bigg\rceil$$ where $\lceil\rceil$ denotes the ceil function and $N_o$ is the length of your actual data.

  You can then derive $N$ as $$N = K(L-r) + r$$ and zero-pad your input data with $N-N_o$ zeros.

  For example, taking your question's requirements, $N_o = 821, L = 256, r = 220$, we get $$K = 17, \quad N = 832$$ and you'd add $N-N_o = 11$ zeros to your input data before computing the spectrogram.

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Long answer

Spectrogram display

1. The number of x-axis indices corresponds to the number of time frames of your spectrogram. $$N_{frames} = \bigg\lfloor\frac{N-r}{L-r}\bigg\rfloor$$ where $N$ is the length of the data, $r$ the overlap and $L$ the window length.

2. Each x-axis index is at the center of a window of length $L$:

$$x_{center}[k] = k \cdot (L-r) + L/2 $$

$k = 0,\dots, N_{frames}-1 $

$\quad \quad$ And the frame edges are $$\left[x_{center}[k]-\frac{L-r}{2},\, x_{center}[k]+\frac{L-r}{2}\right]$$

Verification

For example, taking your values:

• nfft = 256, noverlap = 220: $N_{frames} = \bigg\lfloor\frac{821-220}{256-220}\bigg\rfloor = 16$.

The last frame has center $15\cdot(256-220)+128 = 668$ and edges $$[668-(256-220)/2, 668+(256-220)/2]$$ $$= [650, 686]$$ which you can verify on your Fig2 plot.

• nfft = 256, noverlap = 75: $N_{frames} = 4$

The last frame has center $671$ and edges $$[580.5, 761.5]$$ which you can verify on your Fig3 plot.

• nfft = 256*3, noverlap = 256*2: $N_{frames} = 1$

That's not a short time Fourier transform anymore, just an FFT on 768 data samples. You can see there is only one frame on Fig3. Still, the expected center and edges are verified: center at $384$, edges at $[256, 512]$

Answer 2

If I want to the xaxis set as 821

This doesn't make a lot of sense since you are running afoul of the "uncertainty principle of the Fourier Transform". The shorter a signal is in time, the less frequency resolution has your spectrum.

There are ways to get "one spectrum per sample" but the results are typically not meaningful. Adjacent spectrums will be almost identical and all you are doing is interpolating without gaining any new information.

The easiest way would be set the FFT size to 1, but then each spectrum is just a single value, which is probably not what you want.