## Signal fitting with subsample resolution

**Update**: The most recent version can be found here and a demonstration *here*.

The downloadable code is a "real-life" implementation that is a good choice for "real-life problems", but has grown to almost 700 lines. The code below implements the main functionality only, but is shorter.* *

Determines sub-sample delay and scaling factor between two signals and resamples / scales one signal to match the other.

For example in measurement automation, it is rather common to apply a test signal to a device-under-test and record the output. A delay is caused both from the latency of measurement instruments and the group delay of the device-under-test itself. In some applications, for example when analyzing the distortion properties of power amplifiers, it is necessary to align the signals within a small fraction of a sample length to avoid systematic errors.

The function *fitSignalFFT()* determines the delay with sub-sample resolution. It provides a delayed-and-scaled version of one signal that matches the other one.

This code snippet provides an extended version of the implementation in my "delay estimation by FFT" blog entry.

Resampling a signal with a fractional (subsample) delay assumes reconstruction and resampling of an equivalent continuous-time waveform. There are a few "gotchas" such as "ringing" related to the underlying assumptions (cyclic signal, ideal lowpass as reconstruction filter). Some related background can be found in this code snippet (resampling on a regular grid via FFT) and here (resampling at arbitrary time instants by direct evaluation of the Fourier series).

For non-periodic signals, use adequate zero-padding.

The picture shows an example run:

Two sine waves with different phase and amplitude are used as inputs. The function call returns an offset in samples and a scaling factor that maps the reference signal least-squares-optimally to the other signal.

Further, a delayed and scaled version of the reference signal is returned (green trace, overlapping the blue trace).

Below the example invocation.

*ph = (0:15) * 2 * pi / 16;**ref = sin(ph);**sig = 1.23456 * sin(ph + 0.98765)**[coeff, shiftedRef, delta] = fitSignal_FFT(sig, ref);**figure(); hold on;**plot(ref, 'k+-');**h = plot(sig, 'b+-'); set(h, 'lineWidth', 5);**plot(shiftedRef, 'g+-')**legend('reference signal', 'signal', 'reference shifted / scaled to signal');**title('fitSignal\_FFT demo');*

Copy the following code snippet into a file *fitSignal_FFT.m*.

```
% *******************************************************
% delay-matching between two signals (complex/real-valued)
% M. Nentwig
%
% * matches the continuous-time equivalent waveforms
% of the signal vectors (reconstruction at Nyquist limit =>
% ideal lowpass filter)
% * Signals are considered cyclic. Use arbitrary-length
% zero-padding to turn a one-shot signal into a cyclic one.
%
% * output:
% => coeff: complex scaling factor that scales 'ref' into 'signal'
% => delay 'deltaN' in units of samples (subsample resolution)
% apply both to minimize the least-square residual
% => 'shiftedRef': a shifted and scaled version of 'ref' that
% matches 'signal'
% => (signal - shiftedRef) gives the residual (vector error)
%
% *******************************************************
function [coeff, shiftedRef, deltaN] = fitSignal_FFT(signal, ref)
n=length(signal);
% xyz_FD: Frequency Domain
% xyz_TD: Time Domain
% all references to 'time' and 'frequency' are for illustration only
forceReal = isreal(signal) && isreal(ref);
% *******************************************************
% Calculate the frequency that corresponds to each FFT bin
% [-0.5..0.5[
% *******************************************************
binFreq=(mod(((0:n-1)+floor(n/2)), n)-floor(n/2))/n;
% *******************************************************
% Delay calculation starts:
% Convert to frequency domain...
% *******************************************************
sig_FD = fft(signal);
ref_FD = fft(ref, n);
% *******************************************************
% ... calculate crosscorrelation between
% signal and reference...
% *******************************************************
u=sig_FD .* conj(ref_FD);
if mod(n, 2) == 0
% for an even sized FFT the center bin represents a signal
% [-1 1 -1 1 ...] (subject to interpretation). It cannot be delayed.
% The frequency component is therefore excluded from the calculation.
u(length(u)/2+1)=0;
end
Xcor=abs(ifft(u));
% figure(); plot(abs(Xcor));
% *******************************************************
% Each bin in Xcor corresponds to a given delay in samples.
% The bin with the highest absolute value corresponds to
% the delay where maximum correlation occurs.
% *******************************************************
integerDelay = find(Xcor==max(Xcor));
% (1): in case there are several bitwise identical peaks, use the first one
% Minus one: Delay 0 appears in bin 1
integerDelay=integerDelay(1)-1;
% Fourier transform of a pulse shifted by one sample
rotN = exp(2i*pi*integerDelay .* binFreq);
uDelayPhase = -2*pi*binFreq;
% *******************************************************
% Since the signal was multiplied with the conjugate of the
% reference, the phase is rotated back to 0 degrees in case
% of no delay. Delay appears as linear increase in phase, but
% it has discontinuities.
% Use the known phase (with +/- 1/2 sample accuracy) to
% rotate back the phase. This removes the discontinuities.
% *******************************************************
% figure(); plot(angle(u)); title('phase before rotation');
u=u .* rotN;
% figure(); plot(angle(u)); title('phase after rotation');
% *******************************************************
% Obtain the delay using linear least mean squares fit
% The phase is weighted according to the amplitude.
% This suppresses the error caused by frequencies with
% little power, that may have radically different phase.
% *******************************************************
weight = abs(u);
constRotPhase = 1 .* weight;
uDelayPhase = uDelayPhase .* weight;
ang = angle(u) .* weight;
r = [constRotPhase; uDelayPhase] .' \ ang.'; %linear mean square
%rotPhase=r(1); % constant phase rotation, not used.
% the same will be obtained via the phase of 'coeff' further down
fractionalDelay=r(2);
% *******************************************************
% Finally, the total delay is the sum of integer part and
% fractional part.
% *******************************************************
deltaN = integerDelay + fractionalDelay;
% *******************************************************
% provide shifted and scaled 'ref' signal
% *******************************************************
% this is effectively time-convolution with a unit pulse shifted by deltaN
rotN = exp(-2i*pi*deltaN .* binFreq);
ref_FD = ref_FD .* rotN;
shiftedRef = ifft(ref_FD);
% *******************************************************
% Again, crosscorrelation with the now time-aligned signal
% *******************************************************
coeff=sum(signal .* conj(shiftedRef)) / sum(shiftedRef .* conj(shiftedRef));
shiftedRef=shiftedRef * coeff;
if forceReal
shiftedRef = real(shiftedRef);
end
end
```

## Comments:

First, I would like to thank you for sharing this valuable work. Well, I am working on ultrasonic signals. I have tried to fit the measured signals with a reference one. Due to the fact that time delay and scale are varying, I divided the signals into small pieces where time delay and scale are constant. But, when I reassemble the pieces after fitting, the signal seems to be distorted. In your opinion what is the explanation for this issue ? is there any solution ?

First, I would like to thank you for sharing this valuable work. Well, I am working on ultrasonic signals. I have tried to fit the measured signals with a reference one. Due to the fact that time delay and scale are varying, I divided the signals into small pieces where time delay and scale are constant. But, when I reassemble the pieces after fitting, the signal seems to be distorted. In your opinion what is the explanation for this issue ? is there any solution ?