Detecting Peaks in a FFT Plot

Question

I was wondering how is it possible to detect new peaks within an FFT plot in Python. let's say i have this simple Plot: And i want to automatically measure the 'Similarity' or the Peaks location within a noisy Signal, i have tried to use the cosine Similarity but my real Signal is way too noisy, and with even if i add a new peak to the signal, i keep getting a Cosine of 0.9 since it's only one peak. This is an example of my real signal, and i also have the problem that my signal can be shiffted within the measures, so i can't get a stable frequency array they can be within a window of +/- 100 Hz : This the code that used for the first Plot :

import numpy as np
from pylab import *
import scipy.fftpack

# Number of samplepoints
N = 600
# sample spacing
T = 1.0 / 800.0
x = np.linspace(0.0, N*T, N)
y1 = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x)+ 0.7*np.sin(30.0 * 2.0*np.pi*x)+ 0.5*np.sin(10.0 * 2.0*np.pi*x)
y2 = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x)+ 0.2*np.sin(60.0 * 2.0*np.pi*x)+ 0.4*np.sin(40.0 * 2.0*np.pi*x)
yf1 = scipy.fftpack.fft(y1)
yf2 = scipy.fftpack.fft(y2)
xf = np.linspace(0.0, 1.0/(2.0*T), N/2)

fig, ax = plt.subplots()
plot(xf, 2.0/N * np.abs(yf1[:N/2]))
plot(xf, 2.0/N * np.abs(yf2[:N/2]))
xlabel('Freq (Hz)',fontsize=16,weight='bold')
ylabel('|Y(freq)|',fontsize=16,weight='bold')
ax = gca()
fontsize = 14
for tick in ax.xaxis.get_major_ticks():
    tick.label1.set_fontsize(fontsize)
    tick.label1.set_fontweight('bold')
for tick in ax.yaxis.get_major_ticks():
    tick.label1.set_fontsize(fontsize)
    tick.label1.set_fontweight('bold')
grid(True)
show()
def cosine_similarity(v1,v2):
    "compute cosine similarity of v1 to v2: (v1 dot v2)/{||v1||*||v2||)"
    sumxx, sumxy, sumyy = 0, 0, 0
    for i in range(len(v1)):
        x = v1[i]; y = v2[i]
        sumxx += x*x
        sumyy += y*y
        sumxy += x*y
    return sumxy/math.sqrt(sumxx*sumyy)

print 'Cosine Similarity', cosine_similarity(2.0/N * np.abs(yf1[:N/2]),2.0/N * np.abs(yf2[:N/2]))

I have also though of setting a threshold, but sometime the peaks within the real signal can be smaller than the pre-defined Threshold. Any ideas ?

Answer 1

There are many ways to find peaks, and even to interpolate their sub-sample location. Once you have the peaks, just check if you find a new one.

You can use the peakutils package to find the peaks. You can set there the threshold and minimum distance between peaks.

import numpy as np
from pylab import *
import scipy.fftpack

# Number of samplepoints
N = 600
# sample spacing
T = 1.0 / 800.0
x = np.linspace(0.0, N*T, N)
y1 = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x)+ 0.7*np.sin(30.0 * 2.0*np.pi*x)+ 0.5*np.sin(10.0 * 2.0*np.pi*x)
y2 = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x)+ 0.2*np.sin(60.0 * 2.0*np.pi*x)+ 0.4*np.sin(40.0 * 2.0*np.pi*x)
yf1 = scipy.fftpack.fft(y1)
yf2 = scipy.fftpack.fft(y2)
xf = np.linspace(0.0, 1.0/(2.0*T), N/2)

v1 = 2.0/N * np.abs(yf1[:N/2])
v2 = 2.0/N * np.abs(yf2[:N/2])

# Find peaks
import peakutils
peaks_ind1 = peakutils.indexes(v1, thres=0.2, min_dist=5)
peaks_ind2 = peakutils.indexes(v2, thres=0.2, min_dist=5)

dist_th_for_new_peaks = 3
new_peaks = []
for p in peaks_ind2:
    found_new_peak = np.all(np.abs(p - peaks_ind1) > dist_th_for_new_peaks)
    if found_new_peak:
        new_peaks.append(p)
        print("New Peak!! - %d" % p)

fig, ax = plt.subplots()
plot(xf, v1, color='blue')
plot(xf, v2, color='green')
for p in peaks_ind1:
    ax.scatter(xf[p], v1[p], s=40, marker='s', color='blue', label='v1')
for p in peaks_ind2:
    ax.scatter(xf[p], v2[p], s=40, marker='s', color='green', label='v2')    
for p in new_peaks:
    ax.scatter(xf[p], v2[p], s=40, marker='s', color='red', label='new peaks')        

xlabel('Freq (Hz)',fontsize=16,weight='bold')
ylabel('|Y(freq)|',fontsize=16,weight='bold')

ax = gca()
fontsize = 14
for tick in ax.xaxis.get_major_ticks():
    tick.label1.set_fontsize(fontsize)
    tick.label1.set_fontweight('bold')
for tick in ax.yaxis.get_major_ticks():
    tick.label1.set_fontsize(fontsize)
    tick.label1.set_fontweight('bold')
ax.set_xlim([0,400])
ax.set_ylim([0,0.8])
grid(True)
show()

The red squares are the new peaks that were found in the green signal: