A Signal Subspace Approach for Speech Enhancement

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A Signal Subspace Approach for Speech Enhancement

• Karhunen-Loève Transform Approach
• Sundeep Singh

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Signal Spaces

• Signal occupies an N-dimensional space (for a
  vector of length N)
• For speech signal, the speech occupies only a
  subspace of the entire N-dimensional space
• White noise occupies the entire N-dimensional
  space
• For noisy signal zyw, find the subspace that
  contains the speech and project z onto this
  subspace for the estimate of the clean signal, y

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Subspace Linear Estimator
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Karhunen-Loève Expansion

• Decompose a general second-order random process
  into an orthonormal expansion, whose coefficients
  are uncorrelated random variables
• There exists at least one set of orthonormal
  functions with the property that the coefficients
  in its expansion are uncorrelated random
  variables
• Different from the Fourier series expansion,
  because the coefficients are uncorrelated, or
  statistically orthogonal
• X(t) ?Xnfn(t) (in the mean square sense)

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K-L Transform

• KLT is the theoretically optimal transform for
  transform coding
• Implemented in this algorithm by using an
  empirical estimate based on eigendecomposition of
  the (Toeplitz) covariance matrix
• Decomposes the vector space of the noisy signal
  into a signal(noise) subspace and a noise
  subspace
• Linear estimation is performed by modifying the
  KLT components which lie in the signal subspace
  by a gain function, determined by an estimator
• Enhanced signal is obtained from the inverse KLT
  of the altered components

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K-L Transform (cont.)

• Find covariance matrix of a frame of noisy
  speech, Rz
• Find covariance matrix of the noise, Rw this is
  a diagonal matrix with each element on the
  diagonal equal to the variance of the noise, s²
• s² is estimated from the first few frames of the
  speech signal, since there is all noise and no
  speech

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K-L Transform (cont.)

• Perform eigendecomposition on Rz to find the
  eigenvectors and eigenvalues of the noisy signal,
  RzU?zU
• ?z is a diagonal matrix of the eigenvalues of the
  noisy signal
• U is a matrix of the corresponding eigenvectors
• ?z(k)s² in the noise subspace
• ?z(k)?y(k)s² in the signal subspace
• So for the eigenvalues which are greater than s²,
  we can assume those lie in the signal subspace.

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Implementation

• For each of the eigenvalues, ?z(k), of the noisy
  signal, compare the values to the variance of the
  noise, s²
• If the eigenvalue is greater than the noise
  variance, assume it lies in the signal subspace
  along with its corresponding eigenvector
• Otherwise, assume it lies in the noise subspace,
  and null it and its corresponding eigenvector
• To estimate the eigenvalues, ?y(k), of the clean
  signal, take the noisy eigenvalues which lie in
  the signal subspace and subtract the noise
  variance

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Implementation (cont.)

• The gain function and inverse transform can be
  implemented via a filter, H
• Let yHz be a linear estimator of the clean
  signal y, where H is a NN matrix
• It can be shown that the optimal H to estimate
  the clean signal is HU1GµU1
• U1 is the matrix of eigenvectors that correspond
  to the eigenvalues in the signal subspace
• Gµ is a diagonal matrix, with the gain values as
  elements on the diagonal
• gµ(m) ?y(m) / (?y(m) µs²)
• µ is the LaGrange multiplier, which can be
  estimated in several ways, or chosen to be an
  arbitrary value

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Algorithm Complexity

• Each frame of the noisy signal has a unique
  filter, H, which cleans it up
• Very computationally expensive to implement,
  because a new filter has to be computed for each
  frame
• As frame size increases, the covariance matrices
  get larger, and running time of the algorithm
  increases
• On a 1.3 GHz processor, with 99 CPU utilization
• 5ms frame - 20s running time
• 10ms frame - 48s running time
• 20ms frame - 147s running time

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Audio Demos (µ10)

• Noisy signal, 0 dB SNR
• Enhanced signal
• Noisy signal, 5 dB SNR
• Enhanced signal
• Noisy signal, 10 dB SNR
• Enhanced signal

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Effect of µ on output of algorithm

• As µ increases, the residual noise decreases
  while signal distortion increases
• The effect of changes in µ becomes apparent in
  highly noisy signals
• Noisy signal, -5 db SNR
• Enhanced signal, µ0
• Enhanced signal, µ10
• Enhanced signal, µ100

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The End