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Our system can be described in block diagram form as:
Where:
Assuming the room is an LTI system, y[n] isrelated 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 oftheir 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 determinedby 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].
The room also contains additive noise (which can be recorded). A more accurate block diagram drawing of oursystem is:
The measured recording, y[n] can be related tothe 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 thefrequency 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), sodeconvolution has the undesirable effect of greatly amplifying the noise.
An improvement upon normal deconvolution is to apply a Wiener filter before deconvolution to reduce the additive noise. The Wiener filter utilizes knowledge of thecharacteristics of the additive noise and the signal being recovered to reduce the impact of noise on deconvolution. Thisprocess is known as Wiener deconvolution . The Wiener filter’s mathematical effect on the room’s frequency response can be seenbelow:
Where “x” is the frequency variable, H(x) is the room’s frequency response, G(x) is the wiener-filtered versionof 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 blurringfilter while accounting for noise.
Because of the added S(x) and N(x) terms, Wiener deconvolution cannot be used without knowledge of theoriginal signal and noise. Voice characteristics are fairly predictable, whereas the characteristics of the room are difficultto estimate. Therefore, Wiener deconvolution can only be used when recovering the audio signal (not to determine the roomresponse).
More information on Wiener Deconvolution can be found here .
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