Normalized FFT Difference
The Normalized FFT Difference between two signals provides a point-wise difference between the normalized magnitudes of the Fourier Transforms of the
signals. It allows us to visually compare how the frequency content of the signals differs.
Process:
1. Compute FFT:
• The Fast Fourier Transform (FFT) is computed for both signals to
translate them from the time domain to the frequency domain.
2. Magnitude Calculation:
• The magnitude of the FFT is calculated, which gives the amplitude
of each frequency component in the signal.
3. Normalization:
• The magnitude of the FFT is normalized by dividing it by its maximum value. This brings the values of the magnitude within the range
[0, 1].
4. Difference Calculation:
• The difference between the normalized FFT magnitudes of the two
signals is then calculated at each frequency point.
Mathematical Representation:
If F F T1 (f ) and F F T2 (f ) are the FFTs of signal 1 and signal 2, respectively,
then the normalized FFT difference, F F Tdif f (f ), at frequency f is given by:
F F Tdif f (f ) =
|F F T1 (f )|
|F F T2 (f )|
−
max(|F F T1 (f )|) max(|F F T2 (f )|)
Visualization:
• The Normalized FFT Difference is plotted against the frequency.
• This plot helps in identifying the frequencies at which the signals differ
the most.
1
Interpretation:
• If F F Tdif f (f ) > 0 at a certain frequency, it means that signal 1 has a
higher magnitude at that frequency than signal 2.
• If F F Tdif f (f ) < 0 at a certain frequency, it means that signal 2 has a
higher magnitude at that frequency than signal 1.
• If F F Tdif f (f ) = 0 at a certain frequency, it means that both signals have
the same magnitude at that frequency.
Usage:
• The Normalized FFT Difference is useful in scenarios where we are interested in analyzing how the frequency content of two signals differs.
• It can help in identifying any additional or missing frequency components
in one signal compared to the other.
• It is particularly useful when the signals are expected to be similar, and we
are interested in identifying deviations or anomalies at specific frequencies.
2
Magnitude-Squared Coherence
Magnitude-Squared Coherence provides a frequency-domain measure between 0
and 1 that indicates the linear dependency of two signals at each frequency. It
is used to assess the linear relationship between the two signals in the frequency
domain.
Process:
1. Compute Cross Power Spectral Density:
• The Cross Power Spectral Density (CPSD) between the two signals
is computed, which provides a measure of the linear relationship between the two signals in the frequency domain.
2. Compute Auto Power Spectral Densities:
• The Auto Power Spectral Densities (PSD) of the individual signals
are also computed to normalize the CPSD.
3. Compute Coherence:
• The Coherence is then computed using the CPSD and PSDs.
Mathematical Representation:
The Magnitude-Squared Coherence, Cxy (f ), between two signals x(t) and y(t)
at frequency f is given by:
Cxy (f ) =
|Pxy (f )|2
Pxx (f ) · Pyy (f )
where:
• Pxy (f ) is the Cross Power Spectral Density between signals x(t) and y(t)
at frequency f .
• Pxx (f ) and Pyy (f ) are the Auto Power Spectral Densities of signals x(t)
and y(t) respectively at frequency f .
Visualization:
• The Magnitude-Squared Coherence is plotted against the frequency.
• This plot helps in identifying the frequencies at which the signals are
linearly dependent.
3
Interpretation:
• If Cxy (f ) = 1, it implies perfect linear dependency between the two signals
at frequency f .
• If Cxy (f ) = 0, it implies no linear dependency between the two signals at
frequency f .
• Values between 0 and 1 indicate the degree of linear dependency between
the signals at frequency f .
Usage:
• Magnitude-Squared Coherence is useful in scenarios where we are interested in analyzing how the frequency content of two signals is linearly
related.
• It can help in identifying the frequencies where the signals share common
power, implying a possible linear relationship between them.
4