ECE 8440 Unit Applica,ons (of Homomorphic Deconvolu,on) to Speech Processing

Transcription

1 ECE 8440 Unit Applica,ons (of Homomorphic Deconvolu,on) to Speech Processing Speech produc,on can be modeled as the convolu,on of an excita,on signal with the unit sample response of a linear speech produc,on system. There are two kinds of excita,on signals: 1. pulse train (for "voiced" speech where the vocal cords are vibra,ng) 2. random noise (corresponding to "unvoiced" speech, such as the "s" sound ) This is summarized in the diagram below: Figure Discrete-,me model of speech produc,on

2 Vocal tract model: V(z) = K k =0 P k =0 = AZ K 0 b k z k a k z k k =1 P/2 k =1 K i (1 α k z 1 ) (1 β k z) k =1 The output speech signal is represented by K o (1 r k e jθ k z 1 )(1 r k e jθ k z 1 ) s(n) = v(n) * x(n). equa,on equa,on For voiced speech, x(n) = p(n) = δ(n kn 0 ). k For unvoiced speech, x(n) = r(n) = random noise signal. Using windowing for "short-,me" analysis. Let x(n) = s(n)w(n) be the input to a homomorphic deconvolu,on system where s(n) is a speech signal and w(n) is a window func,on. x(n) = s(n)w(n) For voiced speech, x(n) = [p(n) * v(n)] w(n).

3 If w(n) varies slowly rela,ve to varia,ons of v(n), we can approximate the above as 3 x(n) v(n) * p w (n) where M 1 p w (n) = w(n)p(n) = w(kn 0 )δ(n kn 0 ). k =0 equa,on equa,on Complex cepstrum of x(n) ˆx(n) = ˆv(n) + ˆp w (n) equa,on To obtain ˆp w (n), define a sequence w No (k) = w(kn 0 ), k = 0,1, M-1 0; otherwise equa,on The Fourier Transform of p w (n) is P w (e jω ) = M 1 M 1 p w (k)e jωk M 1 = w(mn 0 )δ(k mn 0 ) e jωk k=0 k=0 m=0 M 1 m=0 M 1 = w(mn 0 ) δ(k mn 0 ) e jωk = w(mn 0 ) e jωmn 0 k=0 M 1 m=0

4 M 1 P w (e jω ) = w(mn 0 ) e jωmn 0 = W N0 (e jωn 0 ) m=0 Period: Set ωn 0 = 2π and solve for ω : ω = 2π log[p w (e jω )] = log[w N0 (e jωn 0 )] equa,on N 0 = period 4 Note that log[p w (e jω )] also has the same period as P w (e jω ). Because of the rela,on between log[p w (e jω )] and log[w N0 (e jωn 0 )],, the rela,on between their inverse DTFT s is ˆp w (n) = ŵ N0 [n / N 0 ] 0 n = ±N 0,±2N 0, otherwise equa,on (Recall this property from material covered in Chapter 4.) Therefore, ˆp w (n) has the form: ˆp w (n) = a i δ(n in 0 ) i

5 As shown before, the complex cepstrum for a signal whose z- transform V(z) has the form shown in equa,on (shown again below) is as follows: 5 = AZ K 0 K i k=1 P/2 k=1 K o (1 α k z 1 ) (1 β k z) k=1 (1 r k e jθ k z 1 )(1 r k e jθ k z 1 ) (equa,on , shown again) ˆv(n) = K 0 n β k, n < 0 k=1 n log A, n = 0 K i n P/2 n α k n + 2r k k=1 n cos(θ n), n > 0. k k=1 (equa,on ) Note: The above expression assumes that the z K 0 in equa,on is removed before calcula,ng the complex cepstrum ˆv(n).

6 Example of Homomorphic Deconvolu,on of Speech Analysis of a sec,on of voiced speech Sampling rate - 8,000 samples per second Hamming window of length 401 used (50 ms window length)

7 Figure Homomorphic deconvolu,on of speech. (a) Segment of speech weighted by a Hamming window. (b) High quefrency component of the signal in (a). (c) Low quefrency component of the signal in (a). 7

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9 Note the two components: One due to v(n) which decreases as 1/n, and the other which is a pulse train due to p(n). 9

10 Figure (a) System for cepstrum analysis of speech signals. (b) Analysis for voiced speech. (c) Analysis for unvoiced speech. 10

11 11 Figure (a) Cepstra and (b) log specta for sequen,al segments of voiced speech.

12 Homomorphic Deconvolu,on of Seismic Signals 12 Two types of seismic signals: 1. Signal generated by an underground explosion and measured ader passing through a por,on of the earth's crust. (Used to detect underground mineral deposits.) 2. Signal generated by a natural underground event (e.g.,, earthquake) Genera,on of the first type of seismic signal can be modeled as shown below: x(n) = s(n) * p(n) where s(n) is a seismic wavelet that depends on the nature of the excita,on and p(n) is the "impulse response" of the por,on of the earth's crust between the excita,on and the point of detec,on. Cepstral analysis can be used to deconvolve s(n) and p(n). Typical signals are shown in the figures below:

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15 Restora,on of Acous,c Recordings 15 A old (low quality) audio recording can be represented as a convolu,on of the true signal with a unit sample response which represented the "distor,ng system," as shown in the figure below: x(n) = s(n) * h(n) Processing approach: es,mate h(n) so that its effect can be compensated by inverse filtering. First, sec,on the available signal into M sec,ons: x m (n) = x(n + mn) n = 0,1,..., N-1 (index within section) m = 0,1,..., M-1 (section index) Assume that Therefore, and x m (n) s m (n) * h(n). X m (e jω ) S m (e jω )H(e jω ) log X m (e jω ) log S m (e jω ) + log H(e jω ). We can obtain an es,mate H e (e jω ) of the frequency response of the distor,ng system H(e jω ) by obtaining averages of and of : log[h e (e jω )] = 1 M M 1 m=0 l og X m (e jω ) log S m (e jω ) log X m (e jω M 1 1 ) - log S M m (e jω ). m=0 log[s(e jω )]

16 1 M M 1 We can obtain by averaging over M data segments. log X m (e jω ) l og X (e jω ) m m= M 1 log S m (e jω ) We can obtain M, which is an es,mate of the "long-,me" power spectrum of m=0 high quality music or singing, from highly quality recordings of music or singing. This averaging is performed on music of the same music type as the music being processed. The desired inverse filter to be used for removing the recording distor,on is then H e 1 (e jω ) = 1 H e (e jω ), Ader es,ma,ng ω < ω p = 0, ω s < ω <π H e 1 (e jω ) as shown above, the remaining processing steps are: 1. Use IDFT of samples of H 1 e (e jω ) to obtain h e 1 (n). 2. Convolve h 1 e (n) with x(n) to "undo" the distor,on introduced by the low- quality recording system.

17 Homomorphic Image Processing We can represent the genera,on of an image using the following rela,on: 17 f(u, v) = f i (u, v)f r (u, v) where f(u, v) f i (u, v) is the image, is an illumina,on func,on, and f r (u, v) is a reflec,on func,on which sa,sfies 0 < f r (u, v) < 1. Therefore, 0 < f(u, v) < f i (u, v) <. Digital images: sampled versions of f(u, v) : f(m,n) = f i (m,n)f r (m,n) Example of processing goals: Use homomorphic system for mul,plica,on to: (a) reduce the dynamic range (for communica,ons and/or storage) (b) enhance contrast (sharpen edges) in the image. f(m,n) ˆf(m,n) ŷ(m,n) y(m,n)

18 (Recall that f(m,n) > 0 so that real log can be used.) 18 ˆf(m,n) = log[fi (m,n)f r (m,n] = log[f i (m,n)] + log[f r (m,n)] = ˆf i (m,n) + ˆf r (m,n) The output of the middle box (linear system) is ŷ(m,n) = L[ˆf(m,n)] = L[ˆf i (m,n) + ˆf r (m,n)] = L[ˆf i (m,n)] + L[ˆf r (m,n)] and the output of the final box (exponen,a,on) is therefore y(m,n) = exp[ŷ(m,n)] { } = exp L[ˆf i (m,n)] + L[ˆf r (m,n)] = exp { L[ˆf i (m,n)]} iexp { L[ˆf r (m,n)]}. Basis for choosing the linear system Illumina,on usually varies "slowly" (gradually) across a scene (although it may vary a large amount over the en,re scene, and therefore have a large overall dynamic range). The reflec,ve component may vary "rapidly," due to sharp edges and changes of texture.

19 Therefore, consider the following linear system for the middle box in the overall system: 19 ŷ(m,n) = γ ˆf(m,n) Then the overall output will be y(m,n) = exp[ŷ(m,n)] = exp[γ ˆf(m,n)] = exp[γ (ˆf i (m,n) + ˆf r (m,n))] = exp[γ (ˆf i (m,n)]exp[γ ˆf r m,n)] y(m,n) = [f i (m,n)] γ [f r (m,n)] γ. To reduce the dynamic range, we need γ < 1. To increase the contrast of the image (increase the ra,o between two different intensi,es), we need γ > 1. However, since f i (m,n) is a "low- frequency" signal and f r (m,n) is a "high frequency" signal, we can use the following linear system to target both goals:

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