Warped Linear Prediction• Concept: Warp the spectrum to emulate human

perception; then perform linear prediction on the result• Approaches to warp the spectrum:

1. Fourier transform, warp, and transform back2. Bank of overlapping band-pass filters. We seen this in

one of the VAD algorithms3. All-pass time-domain filters; all frequencies through

but spectrum and phases are warped• Why? To hopefully be able to more closely model

human speech with smaller residues.• Applications: Speech coding, recognition, synthesis

All-pass filter• A pole of an all-phase filter lies inside the unit

circle and the matching zero is outside. • The magnitudes of the matching poles and

zeros cancel along the unit circle• They lie on the same radius line, so the polar

coordinate angle is the same.• First order all pass filter transfer function:

H(z) = B(z)/A(z) = (z-1 – p0*)/ (1-p0z-1) = (z-1- s e-jφ)/ (1- s ejφz-1) = (z-1- λ)/(1 - λz-1)

• Example: if p0 = ½ + ½i, then the zero is at 1/p0* = 1/(½ - ½ i) = ( ½ + ½ i)/(1/2) = 1 + i

• Higher order all pass filter

p0

p0*

Φ

s

r

an + an-1z-1 + an-2 z-2 + … + a1 z-n+1 + z-n

1 + a1z-1 + a2 z-2 + … + an-1z-n+1 + anz-nH(z) =

Note: p0* = conjugate of p

All pass Filter Visualization

All-pass Filter Phase Response• Real coefficients

– λ, controls the location of the pole (p) and the zero (1/p).

– No phase shift at frequencies 0, π, 2π; only a signal delay

• Complex coefficients– Similar phase responses– Coefficients alter diagonal

crossing frequency: fx

fx = fs/2π arccos(λ) where fs is the sampling rate

– Phase response: w+2arctan(λsin(w)/(1- λcos(w))

π

2π

2π

λ= 0.8

Note: The cross over point is where there is no frequency warping, only a delay

Frequency Warping• All pass filter: magnitude remains constant, but the

phase and frequency warped• Group delay

– Definition: change of phase with respect to change of frequency

– Interpretation: Different frequencies pass through a filter at different speeds. Therefore, a frequency warping operation occurs.

– Formula:

Where w is angle of original frequency, w’ is the angle of the warped frequency, λ is the all-pass coefficient

(1- λ2)sin(w)(1- λ2) cos(w) - 2λ

w’ = arctan

Illustration

Application to LPC

• Warping to the match hearing auditory system– λ = 1.0674(2/π arctan(0.06583 fs/100) ½ -0.1916– Significant at higher sampling rates: > 8k hz– CELP coding:

• Degradation Mean Opinion Score (DMOS): 0.3 < λ < 0.4• Best Bark Scale match: λ = 0.57

• Modified LPC: x’n = d * f; yn ≈ ∑k=1,N ak x’n

– Convolute the frame, f, with all-pass filter, d– Apply linear prediction to warped frequency signal

Evaluation• Extra processing is minimal• The LPC estimate is more accurate than when

warping is not used• For coding operations

– Save one bit per sample at 48 kHz and 32 kHz– Save 0.6 bits per sample at 16kHz– Save 0.3 bits per sample at 8kHz

• Less peaky residue spectrum than standard methods• Insignificant improvement for more than 30 LPC

coefficients

Matlab Toolbox: http://www.acoustics.hut.fi/software/warp

Inverse LPC Filter

• Transfer function: Yz = Hz Xz

– Xz is the original signal

– Hz is the LPC filter ( G / (1-∑i=1,P ai z-i)

– Yz is the filtered signal (residue)

• Inverse filter: Yz / Hz = Xz

– Yz is the filtered output

– Hz is the LPC filter

– Xz is the restored signal

• Convolute the filtered signal with 1/Hz to restore the original signal from the residue

Click Detection using WLP• Definition: A click is a short localized discontinuity

typically less than 1ms, which corrupts a signal• Clicked Detection with both Warped and Standard

linear prediction– LPC: yk = ∑n=1,P an xk-n + rk + ck

– rk is the residue and ck is the energy introduced by clicks

– Looking for spikes (ck), can find click points

• The warped linear prediction coefficient: λ– A value of 0.0 reverts to standard linear prediction– Positive values increase higher frequency resolution– Negative values increase lower frequency resolution

Click Detection Algorithm

• Compute the standard deviation (σ) of the audio signal LPC residue (ex: the amount of residue that we expect to remain)

• FOR each frame– Perform the Linear prediction with various λ values– Consider a click present in the frame when K σ > threshold, where K

is an empirically set gain factor.– Approach 1

• Throw away frames determined to contain clicks• Disadvantage: some distortion is present

– Approach 2• Use interpolation to smooth the residue signal of clicks

• Restore signal: Convolute the inverse LPC filter with the residue

Does WLP have an affect?• Prediction Gain (improvement in signal to noise ratio)

– Divide clean signal energy by residue energy– Note: The residue is computed applying WLP to the noisy signal– The higher the result, the better the detection– Gp = 10 log (∑n=1,N |xn|2 / ∑n=1,n |rn|2)

• Experiment– 44 kHz sample rate, 215 frames of 1024 samples, musical signal

corrupted with known click points, λ values varied between -0.8 and +0.8

– Result: choice of λ affects the ratio between clean signal and residue with clicks

λ -0.8 -0.4 0.0 0.4 0.8

Gp 35.51 27.49 22.37 17.30 11.04

Experiment• Approach 1: Throw away click frames

• Approach 2: Interpolate click frames

• Results: Both LPC and WLP can detect clicks WLP with warping coefficient -0.7 reduces false detects LPC and WLP miss approximately the same number of clicks