How to Use the Connexions Document Template

Our system can be described in block diagram form as:

Figure 1

Where:

Assuming the room is an LTI system, y[n] is related to f[n] and h[n] by discrete time convolution:

f[n]*h[n]=y[n]

Convolution is commutative so the following also holds:

h[n] * f[n] = y[n]

Taking the Discrete Time Fourier Transform of f[n], h[n], and y[n] shows that in the frequency domain, the convolution of f[n] and h[n] is equivalent to multiplication of their Fourier counterparts:

F(jw) H(jw) = Y(jw)

Given a known original signal and a known measured recording, the room’s frequency response can be determined by division in the frequency domain:

H(jw) = Y(jw) /F(jw)

Similarly, given a known room response and known measured recording, the original signal can be determined by division in the frequency domain.

F(jw) = Y(jw) /H(jw)

The inverse DTFT can then be used to determine the impulse response h[n] or the recovered signal f[n].

Room Noise

The room also contains additive noise (which can be recorded). A more accurate block diagram drawing of our system is:

Figure 2

The measured recording, y[n] can be related to the original signal, room response, and noise in frequency as:

F(jw) H(jw) + N(jw)= Y(jw)

In order to compute the room’s frequency response or the DTFT of the recovered signal, division in the frequency domain is again performed:

H(jw) = (Y(jw) / F(jw)) – (N(jw)/ F(jw))

F(jw) = (Y(jw) / H(jw)) – (N(jw)/ H(jw))

Many of the fourier coefficients of the room response are small (especially at high frequencies), so deconvolution has the undesirable effect of greatly amplifying the noise.

Noise Reduction

An improvement upon normal deconvolution is to apply a Wiener filterbefore deconvolution to reduce the additive noise. The Wiener filter utilizes knowledge of the characteristics of the additive noise and the signal being recovered to reduce the impact of noise on deconvolution. This process is known as Wiener deconvolution. The Wiener filter’s mathematical effect on the room’s frequency response can be seen below:

Where “x” is the frequency variable, H(x) is the room’s frequency response, G(x) is the wiener-filtered version of the inverse of the room response and, S(x) is the expected signal strength of the original signal f[n], and N(x) is the expected signal strength of the additive noise.

F(x) =G(x) Y(x)

Where F(x) is the DTFT of the recovered signal and Y(x) is the DTFT of the measured recording.

The following example from image processing shows effectiveness of Wiener deconvolution at reversing a blurring filter while accounting for noise.

Figure 3

Because of the added S(x) and N(x) terms, Wiener deconvolution cannot be used without knowledge of the original signal and noise. Voice characteristics are fairly predictable, whereas the characteristics of the room are difficult to estimate. Therefore, Wiener deconvolution can only be used when recovering the audio signal (not to determine the room response).

More information on Wiener Deconvolution can be found here.