How a spectrum is computed in PyRPL

Inspiration comes from Oppenheim & Schaefer 1975 and from Agilent

The spectrum analyzer in Pyrpl estimates the spectrum of internal or external signals by performing Fast-Fourier Transforms of traces recorded by the scope. Since in the current version of Pyrpl, the stream of data from the scope is made of discontiguous segments of 2^14 samples, we are currently using the Bartlett method, which consists in the following steps:

1. Each segment is multiplied by a symmetric window function of the same size.
2. the DFT of individual segments is performed.
3. The square modulus of the resulting periodograms are averaged to give the estimate of the spectrum, with the same size as the initial time-segments.

A variant of this method is the Welch method, in which the segments are allowed to be overlapping with each other. The advantage is that when a narrow windowing function (ie a large number of “points-per-bandwidth” in the frequency domain) is used, the points far from the center of the time-segments have basically no weight in the result. With overlapping segments, it is basically possible to move the meaningful part of the window over all the available data. This is the basic principle of real-time spectrum analyzers. This cannot be implemented “as is” since the longest adjacent time-traces at our disposal is 2^14 sample long.

However, a possible improvement, which would not require any changes of the underlying FPGA code would be to apply the welch method with subsegments smaller than the initial scope traces: for instance we would extract 2^13 points subsegments, and we could shift the subsegment by up to 2^13 points. With such a method, even with an infinitely narrow windowing function, we would only “loose” half of the acquired data. This could be immediately implemented with the Welch method implemented in scipy.

In the following, we discuss the normalization of windowing functions, and then, the basic principle of operation of the two modes “iq” and “baseband”.

Normalization of windowing functions

The Fourier transform of the series a_n is defined by

A_m = 1/N sum(a_k exp(-2 i pi m k/N)) [1]

With this convention, we need to pay attention that the DC-component is for m=0, and the “negative frequencies” are actually located in the second half of the interval [N/2, N]. Indeed, we can show that because of the discretization, A_{N - m} = A_{-m}

We can also show that the Fourier transform of the product of time series a_n and windowing function f_n, is the convolution of their Fourier Transform:

FT(a_n f_n)_m = A_m * F_m [2]

Let’s first consider the case of a pure sinusoid a_n = cos(2 pi l n/N). The Fourier transform is A_m = (delta(l-m) + delta(l+m))/2. Hence, the Fourier transform is given by

FT(a_n f_n) = (F_(N-l) + F(l))/2 [3]

Moreover, a reasonable windowing function will only have non-zero Fourier components on the few bins around DC, such that if we measure a pure sinusoid with a frequency far from 0, there wont be any significant overlap between the two terms, and we will measure 2 distinct peaks in the positive and negative frequency regions, each of them with the shape of the Fourier transform of the windowing function.

Normalization for coherent signals

If we want the maximum value of the peaks to correspond to the amplitude of the sine (that is to say, a correct normalization in terms of Vpk), we need to make sure that the peak value of F_m is 2. Since the maximum of the windowing function is the DC-component F_0, we need to take:

F_0 = 2 [4]

or equivalently

1/N sum(f_n) = 2 [5]

To be complete, if the sinusoid has a frequency close to 0, the two terms in [3] take significant values for the same values of frequency, and they will interfere with each other (since they are complex numbers). A consequence of that is that the phase of the signal, which reflects into the complex arguments of F_(N-l) and F(l) starts to play a significant role, and thus, the spectrum is not stationary anymore, but oscillates in time with the frequency of the signal. The oscillations are most visible on the DC bin, where they oscillate between 0 and F(0), with the dependence:

|FT[a_n f_n]|^2 = 4 sin^2(2 pi l/N) [6]

but the oscillation is also present, with a reduced contrast on the neighboring frequency bins. There is not much we can do about it, except maybe to correct to make sure the average value of the oscillations is 1 (instead of 2 in formula [6], due to the fact that negative and positive frequency components both contribute to the averaged spectrum in this regime).

Normalization for noise spectral density:

On the other hand, let’s consider a white Gaussian noise with variance 1. From Wiener Khintchine theorem, it should correspond to a flat spectrum of value 2 pi. From equation [2], since the variance of A_m fulfills Wiener Khintchine theorem, we deduce that the windowing function should fulfill:

sum(F_m^2) = 1 [7]

Using Parseval’s theorem, this is equivalent to

sum(f_n^2) = 1 [8]

By comparing eq [8] and eq [5], we arrive at the interesting conclusion that the windowing function should be normalized with a linear summation for coherent signal measurements in Vpk^2 and with a quadratic summation for power spectral densities in Vpk^2/Hz. This is the time-domain counterpart of the fact that coherent signals are only sensitive to a single value of F_m, while noise spectra are integrated over the whole spectrum of the filtering window (eq [4] and [7]).

In pyrpl, we actually decided to make sure both conditions are fulfilled simultaneously by defining the rbw of a given filtering window to be:

rbw = sum(f_n^2)/sum(f_n) [9]

With this choice, the correct results are retrieved if we make all calculations in Vpk^2, and divide the results by the rbw to convert them in Vpk^2/Hz.

For this reason, the rbw is not exactly the width at 3 dB of the filter spectrum, but actually depends on the precise shape of the window over the whole frequency range via eq [9].

IQ mode

In iq mode, the signal to measure is fed inside an iq module, and thus, multiplied by two sinusoids in quadrature with each other, and at the frequency center_freq. The resulting I and Q signals are then filtered by 4 first order filters in series with each other, with cutoff frequencies given by span. Finally, these signals are measured simultaneously with the 2 channels of the scope, and we form the complex time serie c_n = I_n + i Q_n. The procedure described above is applied to extract the periodogram from the complex time-serie.

Since the data are complex, there are as many independent values in the FFT than in the initial data (in other words, negative frequencies are not redundant with positive frequency). In fact, the result is an estimation of the spectrum in the interval [center_freq - span/2, center_freq + span/2].

Baseband

In baseband mode, the signal to measure is directly fed to the scope and the procedure described above is applied directly. There are 2 consequences of the fact that the data are real:

1. The negative frequency components are complex conjugated (and thus redundant) wrt the positive ones. We thus throw away the negative frequencies, and only get a measurement on the interval [0, span/2]
2. The second scope channel can be used to measure another signal.

It is very interesting to measure simultaneously 2 signals, because we can look for correlations between them. In the frequency domains, these correlations are most easily represented by the cross-spectrum. We estimate the cross-spectrum by performing the product conjugate(fft1)*fft2, where fft1 and fft2 are the DFTs of the individual scope channels before taking their modulus square.

Hence, in baseband mode, the method curve() returns a 4x2^13 array with the following content: - spectrum1 - spectrum2 - real part of cross spectrum - imaginary part of cross spectrum

Proposal for a cleaner interface for spectrum analyzer:

To avoid baseband/2-channels acquisition from becoming a big mess, I suggest the following:

• The return type of the method curve should depend as little as possible from the particular settings of the instrument (channel2_baseband_active, display_units). That was the idea with scope, and I think that makes things much cleaner. Unfortunately, for baseband, making 2 parallel piplines such as curve_iq, curve_baseband is not so trivial, because curve() is already part of the AcquisitionModule. So I think we will have to live with the fact that curve() returns 2 different kinds of data in baseband and iq-mode.
• Moreover, in baseband, we clearly want both individual spectra + cross-spectrum to be calculated from the beginning, since once the abs() of the ffts is taken, it is already too late to compute conjugate(fft1)*fft2
• Finally, I suggest to return all spectra with only one “internal unit” which would be V_pk^2: indeed, contrary to rms-values unittesting doesn’t require any conversion with peak values, moreover, averaging is straightforward with a quadratic unit, finally, .../Hz requires a conversion-factor involving the bandwidth for unittesting with coherent signals

I suggest the following return values for curve():

• In normal (iq-mode): curve() returns a real valued 1D-array with the normal spectrum in V_pk^2
• In baseband: curve() returns a 4xN/2-real valued array with (spectrum1, spectrum2, cross_spectrum_real, cross_spectrum_imag). Otherwise, manipulating a complex array for the 2 real spectra is painful and inefficient.

Leo: Seems okay to me. One can always add functions like spectrum1() or cross_spectrum_complex() which will take at most two lines. Same for the units, I won’t insist on rms, its just a matter of multiplying sqrt(1/2). However, I suggest that we then have 3-4 buttons in the gui to select which spectra and cross-spectra are displayed.

Yes, I am actually working on the gui right now: There will be a baseband-area, where one can choose display_input1_baseband, input1_baseband, display_input2_baseband, input2_baseband, display_cross_spectrum, ‘display_cross_spectrum_phase’. And a “iq-area” where one can choose center_frequency and input. I guess this is no problem if we have the 3 distinct attributes input, input1_baseband and input2_baseband, it makes thing more symmetric...

IQ mode with proper anti-aliasing filter

When the IQ mode is used, a part of the broadband spectrum of the two quadratures is to be sampled at a significantly reduced sampling rate in order to increase the number of points in the spectrum, and thereby resolution bandwidth. Aliasing occurs if significant signals above the scope sampling rate are thereby under-sampled by the scope, and results in ghost peaks in the spectrum. The ordinary way to get rid of this effect is to use excessive digital low-pass filtering with cutoff frequencies slightly below the scope sampling rate, such that any peaks outside the band of interest will be rounded off to zero. The following code implements the design of such a low-pass filter (we choose an elliptical filter for maximum steepness):

import numpy as np
from scipy import signal
import matplotlib.pyplot as plt

# the overall decimation value
decimation = 8

# elliptical filter runs at ell_factor times the decimated scope sampling rate
ell_factor = 4

wp = 0.8/ell_factor # passband ends at xx% of nyquist frequency
ws = 1.0/ell_factor # stopband starts at yy% of nyquist frequency
gpass = 5. # jitter in passband (dB)
gstop = 20.*np.log10(2**14)  # attenuation in stopband (dB)
#gstop = 60  #60 dB attenuation would only require a 6th order filter
N, Wn = signal.ellipord(wp=wp, ws=ws, gpass=gpass, gstop=gstop, analog=False)  # get filter order
z, p, k = signal.ellip(N, gpass, gstop, Wn, 'low', False, output='zpk')  # get coefficients for implementation
b, a = signal.ellip(N, gpass, gstop, Wn, 'low', False, output='ba')  # get coefficients for plotting
w, h = signal.freqz(b, a, worN=2**16)
ww = np.pi / 62.5  # scale factor for frequency axis (original frequency axis goes up to 2 pi)

# extent w to see what happens at higher frequencies
w = np.linspace(0, np.pi, decimation/ell_factor*2**16, endpoint=False)
# fold the response of the elliptical filter
hext = []
for i in range(decimation/ell_factor):
    if i%2 ==0:
        hext += list(h)
    else:
        hext += reversed(list(h))
h = np.array(hext)
# elliptical filter
h_abs = 20 * np.log10(abs(h))

# 4th order lowpass filter after IQ block with cutoff of decimated scope sampling rate
cutoff = np.pi/decimation
butter = 1.0/(1.+1j*w/cutoff)**4
butter_abs = 20 * np.log10(abs(butter))

# moving average decimation filter
M = float(decimation) # moving average filter length
mavg = np.sin(w*float(M)/2.0)/(sin(w/2.0)*float(M))
mavg_abs = 20 * np.log10(abs(mavg))

# plot everything together and individual parts
h_tot = h_abs + mavg_abs + butter_abs
plt.plot(w/ww, h_tot, label="all")
plt.plot(w/ww, h_abs, label="elliptic filter")
plt.plot(w/ww, butter_abs, label="butterworth filter")
plt.plot(w/ww, mavg_abs, label="moving average filter")


plt.title('Elliptical lowpass filter of order %d, decimation %d, ell_factor %d'%(N, decimation, ell_factor))
plt.xlabel('Frequency (MHz)')
plt.ylabel('Amplitude (dB)')
plt.grid(which='both', axis='both')
plt.fill([ws/ww*np.pi/decimation*ell_factor, max(w/ww), max(w/ww), ws*np.pi/ww/decimation*ell_factor], [max(h_abs), max(h_abs), -gstop, -gstop], '0.9', lw=0) # stop
plt.fill([wp/ww*np.pi/decimation*ell_factor, min(w/ww), min(w/ww), wp*np.pi/ww/decimation*ell_factor], [min(h_abs), min(h_abs), -gpass, -gpass], '0.9', lw=0) # stop
plt.axis([min(w/ww), max(w/ww), min(h_abs)-5, max(h_abs)+5])
plt.legend()
plt.show()
plt.savefig('c://lneuhaus//github//pyrpl//doc//specan_filter.png',DPI=300)

print "Final biquad coefficients [b0, b1, b2, a0, a1, a2]:"
for biquad in signal.zpk2sos(z, p, k):
    print biquad

Resulting filter

We see that a filter of 8th order, consisting of 4 sequential biquads is required. Since we do not require the span / sampling rate of the spectrum analyzer to be above roughly 5 MHz, we may implement the four biquads sequentially. Furthermore, for even lower values of the span, the filter can be fed with a reduced clock rate equal to the scope decimation factor divided by the variable ‘decimation’ in the filter design code above (4 in the example). For the aliasing of the lowpass filter passband not to cause problems in this case, we must in addition use the 4th order butterworth lowpass already available from the IQ module and the moving average filter of the scope. Then, as the plot shows, we can be sure that no aliasing occurs, given that no aliasing from the ADCs is present (should be guaranteed by analog Red Pitaya design).

The problem with our scheme is the complexity of introducing 2 (for the two quadratures) 4-fold biquads. This will not fit into the current design and must therefore be postponed to after the FPGA cleanup.

We could however opt for another temporary option, applicable only to stationary signals: Measure the spectrum twice or thrice with slightly shifted IQ demodulation frequency (at +- 10% of span and the actual center, as required above), and only plot the pointwise-minimum (with respect to the final frequency axis) of the obtained traces. This is simple and should be very effective (also to reduce the central peak at the demodulation freuqency), so i suggest we give it a try. Furthermore, it prepares the user that IQ spectra will only have 80% of the points in baseband mode, which will remain so after the implementation of the lowpass filter. The plot above shows that we do not have to worry about aliasing from multiple spans away if the bandwidth if the IQ module is se to the scope sampling rate (or slightly below). I am not aware that this method is used anywhere else, but do not see any serious problem with it.