Big FFT amplitude difference between the existing

Question

I think, what amplitude in the noised regions on the FFT-plot of the filtered signal should be lower, then now. Probably, small numeric deviations/errors presented in the scipy.fftpack.lfilter().

I tried to add noise to the existing signal, but no result.

Why FFT-amplitude of the filtered signal (green) in the noise regions are so high?

Update:
300 dB of an FFT-amplitude are non-physical - it's clear, it's due to 64bit float in the Python-environment.

FFT of the filtered signal (green) has so low dB (as ~67 dB) due to ~4000 samples at all signal (not 'per second' as on the picture, little mistake, not critical), sample rate = 200 samples/sec. 1 frequency bin=200/4000/2=0.025Hz, 2000 bins shown.

If we take a longer signal, we get a more frequency resolution per bin (i.e. 40 000samples, 1 freq bin = 200/40 000/2 = 0.0025 Hz). And also we get FFT of a filtered signal ~87 dB.

(Numbers 67 dB and 87 dB seems to be non-physical, because of an initial signal SNR 300dB, but I try to add some noise to the existing signal and get the same numbers)

If you want to get a picture of a non-depend FFT to the number of samples in a signal, you should use a windowed FFT and slide window to compute a whole-signal-FFT.

'''
Created on 13.02.2013, python 2.7.3

@author: 
'''
from numpy.random import normal
from numpy import sin, pi, absolute, arange, round
#from numpy.fft import fft
from scipy.fftpack import fft, ifft
from scipy.signal import kaiserord, firwin, lfilter, freqz
from pylab import figure, clf, plot, xlabel, ylabel, xlim, ylim, title, grid, axis, show, log10,\
                  subplots_adjust, subplot

def filter_apply(filename):
    pass

def sin_generator(freq_hz = 1000, sample_rate = 8000, amplitude = 1.0, time_s = 1):
    nsamples = round(sample_rate * time_s)
    t = arange(nsamples) / float(sample_rate.__float__())
    signal = amplitude * sin(2*pi*freq_hz.__float__()*t)
    return signal, nsamples

def do_fir(signal, sample_rate):
    return signal

#-----------------make a signal---------------
freq_hz = 10.0
sample_rate = 400
amplitude = 1.0
time_s = 10
a1, nsamples = sin_generator(freq_hz, sample_rate, amplitude, time_s)
a2, nsamples = sin_generator(50.0, sample_rate, 0.5*amplitude, time_s)
a3, nsamples = sin_generator(150.0, sample_rate, 0.5*amplitude, time_s)
mu, sigma = 0, 0.1 # mean and standard deviation
noise = normal(mu, sigma, nsamples)
signal = a1 + a2 + a3 # + noise

#----------------create low-pass FIR----
# The Nyquist rate of the signal.
nyq_rate = sample_rate / 2.0
# The desired width of the transition from pass to stop,
# relative to the Nyquist rate.  We'll design the filter
# with a 5 Hz transition width.
width = 5.0/nyq_rate

# The desired attenuation in the stop band, in dB.
ripple_db = 60.0

# Compute the order and Kaiser parameter for the FIR filter.
N, beta = kaiserord(ripple_db, width)
print 'N = ',N, 'beta = kaiser param = ', beta

# The cutoff frequency of the filter.
cutoff_hz = 30.0

# Use firwin with a Kaiser window to create a lowpass FIR filter.
# Length of the filter (number of coefficients, i.e. the filter order + 1)
taps = firwin(N, cutoff_hz/nyq_rate, window=('kaiser', beta))

# Use lfilter to filter x with the FIR filter.
filtered_signal = lfilter(taps, 1.0, signal)

#----------------plot signal----------------------
hh,ww=2,2  
figure(figsize=(12,9))  
subplots_adjust(hspace=.5)
#figure(1)
subplot(hh,ww,1)

# existing signal
x = arange(nsamples) / float(sample_rate)

# The phase delay of the filtered signal.
delay = 0.5 * (N-1) / sample_rate

# original signal
plot(x, signal, '-bo' , linewidth=2) 

# filtered signal shifted to compensate for 
# the phase delay.
plot(x-delay, filtered_signal, 'r-' , linewidth=1)

# Plot just the "good" part of the filtered signal. 
# The first N-1 samples are "corrupted" by the
# initial conditions.
plot(x[N-1:]-delay, filtered_signal[N-1:], 'g', linewidth=2)

xlabel('time (s)')
ylabel('amplitude')
axis([0, 1.0/freq_hz*2, -(amplitude*1.5),amplitude*1.5]) # two periods of freq_hz
title('Signal (%d samples)' % nsamples)
grid(True)


#-------------- FFT of the signal
subplot(hh,ww,2)

signal_fft=fft(signal)
filtered_fft =fft(filtered_signal[N-1:])

# existing signal 
y = 20*log10( ( abs( signal_fft/nsamples )*2.0)/max( abs( signal_fft/nsamples )*2.0) )# dB Amplitude
x = arange(nsamples)/float(nsamples)*float(sample_rate)

# filtered signal
y_filtered = 20*log10( (abs(filtered_fft/ (nsamples - N + 1) )*2.0)/max(abs(signal_fft/ (nsamples - N + 1) )*2.0) )# dB Amplitude
x_filtered = arange(nsamples - N + 1)/float(nsamples - N + 1)*float(sample_rate)
yy = fft(ifft(filtered_fft))

plot(x,y, linewidth=1) 
plot(x_filtered, y_filtered, 'g', linewidth=2)
xlim(0, sample_rate/2) # compensation of mirror (FFT imaginary part)
xlabel('freq (Hz)')
ylabel('amplitude, (dB)')
title('Signal (%d samples)' % nsamples)
grid(True)


#--------------FIR ampitude response
subplot(hh,ww,3)

w, h = freqz(taps, worN=8000)
#plot((w/pi)*nyq_rate, absolute(h), linewidth=2)
plot((w/pi)*nyq_rate, 20*log10(absolute(h)/1.0),'r', linewidth=1)
xlabel('Frequency (Hz)')
#ylabel('Gain -blue')
ylabel('Gain (dB)')
title('Frequency Response')
#ylim(-0.05, 1.05)
grid(True)


#--------------FIR coeffs
subplot(hh,ww,4)
plot(taps, 'bo-', linewidth=2)
title('Filter Coefficients (%d taps)' % N)
grid(True)

show()

Answer 1

I think you -300dB of noise is pretty low. Keep in mind the this is a log scale, so we are talking about a couple of digits at 64bit resolution.

Using double precision float (64bit) you won't get any lower.