Extracting the system impulse response from MLS measurement

The frequency content of the signal recorded at the microphone will contain information on the frequency response of the system under test. In order to go beyond this and get the impulse response of the system, we must use the auto-correlation property of the MLS signals. First recognise that the measured signal is the convolution of the MLS and the system impulse response. We will define the input impulse response of our system as $h$. We will refer to $h$ as the system impulse response to prevent confusion with the input impulse response of a pulse reflectometry test object. This distinction will be discussed in more detail in section 7.5.4. The MLS signal is $s$, so the pressure, $y$, measured at the microphone will be

$$y = s * h$$ (7.8)

where $*$ here denotes convolution. Performing correlation with respect to $s$ on both sides of equation (7.8) gives [70]:

$$\phi_{sy} = \phi_{ss} * h = \rho * h$$ (7.9)

where $\phi_{ab}$ is notation for the correlation of $a$ and $b$. Note that convolution in the time domain is multiplication in the frequency domain, so the fact that the frequency spectrum of $\rho$ is flat, except for the zero frequency component, means that $h$ is left unchanged by convolution with $\rho$ except for a small dc offset of the order of $1/n$. The impulse response of the system can therefore be extracted from the measurement of the system response by correlation with the MLS input. Correlation is defined as

$$\phi_{sy}(i) = \frac{1}{n} \sum\limits_{j=0}^{n-1} s_j y_{i+j}^* = \frac{1}{n} \sum\limits_{j=0}^{n-1} s_{j-i} y_{j}^*$$ (7.10)

which can be converted to a matrix notation by making a matrix $S_n$ consisting of $n$ circularly shifted versions of $s$ [70]:

$${\mathbf \Phi}_{sy} = \frac{1}{n} S_n {\mathbf Y}.$$ (7.11)

The elements of $S_n$ are given by $S_n(i,j) = s_{j-i}$ where $j-i$ is taken modulo $n$ so that the successive rows of the matrix contain $s$ shifted one step to the right each time with the values leaving on the right appearing on the left. ${\mathbf Y}$ is a column vector of the measured system response and ${\mathbf \Phi}_{sy}$ a column vector of the resulting correlation.

The cross-correlation can also be performed in the frequency domain by considering the close relationship between cross-correlation and convolution. Deconvolution was performed by frequency domain division in chapter 5. Convolution on the other hand may performed by multiplication in the frequency domain. Cross-correlation of two signals is the reverse of the first sequence convolved with the second sequence [59] pp.92-96:

$$\phi_{ab}(t) = a(-t) * b(t).$$ (7.12)

Reversal in the time domain means complex conjugation in the frequency domain:

$$\mathrm{fft}(a(-t)) = (\mathrm{fft}(a(t)))^*.$$ (7.13)

It therefore follows that the cross-correlation of two signals in the time domain becomes the conjugate of the first signal multiplied by the second signal in the frequency domain.

$$\mathrm{fft}(\phi_{ab}) = \frac{1}{n}(\mathrm{fft}(a))^* \times (\mathrm{fft}(b)).$$ (7.14)

Discrete Fourier transforms are used for the analysis in this chapter. The speed of analysis is acceptable for the measurements we present here. Before acceptable computational power was available, it was necessary to perform interpolation to make the length of the sequence up to $2^m$, enabling the use of fast Fourier transforms [66]. Another option is the fast Hadamard transform technique as set out in Borish and Angell [70] which does not require interpolation and is less computationally expensive.