如何使用np.fft.fft制作PSD图？

Question

I want to make a plot of power spectral density versus frequency for a signal using the numpy.fft.fft function. I want to do this so that I can preserve the complex information in the transform and know what I'm doing, as apposed to relying on higher-level functions provided by numpy (like the periodogram function). I'm following Mathwork's nice page about doing PSD analysis using Matlab's fft function: https://www.mathworks.com/help/matlab/ref/fft.html

In this example, I expect the PSD to peak at the frequency I used to construct the signal, which was 100 in this case. I generate the signal using 1000 time points a frequency of 100 inverse time units. I thought that the fft magnitude could be plotted against [0, nt/2] and the peaks would show up where there is the most energy in the frequency. When I did this, things went wrong. I expected my PSD to peak at 100.

How can I make a spectral density plot of frequency vs energy contained in that frequency using np.fft.fft ?

Edit

to clarify, in my real problem, I only know that my characteristic frequency is much larger than my sample frequency

import matplotlib.pyplot as plt
import numpy as np
t = np.arange(1000)
sp = np.fft.fft(np.sin(100 * t * np.pi))
trange = np.linspace(0, t[-1] / 2, t.size)
plt.plot(trange, np.abs(sp) / t.size)
plt.show()

This is a sketch I made of the expected output:

Answer 1

What is your sample frequency? This sequence you are generating can represent a infinite number of continuous time signals according to the sample frequency. The sample frequency needs to be at least twice the maximum signal frequency, as stated by the Sampling Theorem, so, using fs = 250Hz and using a sine of 10 seconds it becomes:

import matplotlib.pyplot as plt
import numpy as np

fs = 250
t = np.arange(0, 10, 1/fs)
sp = np.fft.fft(np.sin(2*np.pi * 100 * t))
trange = np.linspace(0, fs, len(t))
plt.plot(trange, np.abs(sp))
plt.show()

If you run this you will see a peak at 100Hz as expected.