74
J. Kumar V. and K. A. Reddy
each row of X is made of n (period of the input) samples as given below,
⎡
⎢
⎢
⎢
⎣
x(1)
x(2)
. . . x(n)
x(n + 1)
x(n + 2)
. . . x(2n)
. . .
. . .
. . .
x((m − 1)n + 1) x((m − 1)n + 2) . . . x(mn)
⎤
⎥
⎥
⎥
⎦
(3.35)
Applying SVD to X we get the diagonal matrix S to be a rank one matrix with
the first entry alone nonzero and all other entries (S 2 to S n ) will be zero. Rephrasing,
we can say that if samples of a periodic signal are formed into a matrix X such that
the rows contain samples of an integral number of periods of the signal, then the
dominant first singular valueS 1 obtained after applying SVD to matrix X indicates
a strong periodic component in the rows of the data matrix. It can also be shown
that the periodic signal can be recovered by computing s 1 u 2 v T2 . Here u 1 and v 1 are
column matrix extracted from the first columns of the corresponding U and V (called
left singular vector and right singular vector). If noise is present in the signal or if the
signal is quasi-periodic, then after applying SVD we will get nonzero values for to
S 2 to S n . However, if we obtain a matrix X r with a row length r containing samples
of integral number of periods of the signal or the dominant frequency (principal
component) of a quasiperiodic signal, then the ratio of first two singular values
S 1r /S 2r for X r will be a maximum.
Using this artefact-free PPG can be extracted from a motion artefact corrupted
PPG. The steps involved are:
(1) Take M samples of the data, say x(k), k = 1…M of the PPG signal to be
processed.
(2) Prepare different matrices X 1 , X 2 …X r …X n , of different row lengths using x(k)
such that the row lengths are in the range 0.8 Hz to 2 Hz (the dominant frequency
PPG will be in this range)
(3) Perform SVD on each of these matrices X 1 , X 2 …X r …X n
(4) For each SVD, compute the singular value ratio (SVR) of the first two singular
values. Here SVR = .s 1 /s 2
(5) Plot the SVR against the row length r to obtain a graph called the SVR spectrum
of the signal.
(6) From the SVR spectrum, determine the value of row length, say r, for which the
SVR (s 1r /s 2r ) is maximum as seen in a typical SVR spectrum shown in Fig. 3.9.
(7) From the SVD of the corresponding data matrix X r reconstruct the PPG signal
by taking the average of all rows of s 1 u 1 v 1
t .
It should be noted here that the averaging process eliminates not only artefact
present in the PPG but also eliminates any additive noise in the PPG. Once one cycle
of a PPG is recovered, a kind of roll over procedure can be implemented by removing
the first r samples from and appending r new samples at the end.
For example, if the initial samples are x(k), k = 1…M then, after extracting the
first PPG cycle, x(k) is modified as: x(k), k = (r + 1) … (M + r). Once again steps 2
J. Kumar V. and K. A. Reddy
each row of X is made of n (period of the input) samples as given below,
⎡
⎢
⎢
⎢
⎣
x(1)
x(2)
. . . x(n)
x(n + 1)
x(n + 2)
. . . x(2n)
. . .
. . .
. . .
x((m − 1)n + 1) x((m − 1)n + 2) . . . x(mn)
⎤
⎥
⎥
⎥
⎦
(3.35)
Applying SVD to X we get the diagonal matrix S to be a rank one matrix with
the first entry alone nonzero and all other entries (S 2 to S n ) will be zero. Rephrasing,
we can say that if samples of a periodic signal are formed into a matrix X such that
the rows contain samples of an integral number of periods of the signal, then the
dominant first singular valueS 1 obtained after applying SVD to matrix X indicates
a strong periodic component in the rows of the data matrix. It can also be shown
that the periodic signal can be recovered by computing s 1 u 2 v T2 . Here u 1 and v 1 are
column matrix extracted from the first columns of the corresponding U and V (called
left singular vector and right singular vector). If noise is present in the signal or if the
signal is quasi-periodic, then after applying SVD we will get nonzero values for to
S 2 to S n . However, if we obtain a matrix X r with a row length r containing samples
of integral number of periods of the signal or the dominant frequency (principal
component) of a quasiperiodic signal, then the ratio of first two singular values
S 1r /S 2r for X r will be a maximum.
Using this artefact-free PPG can be extracted from a motion artefact corrupted
PPG. The steps involved are:
(1) Take M samples of the data, say x(k), k = 1…M of the PPG signal to be
processed.
(2) Prepare different matrices X 1 , X 2 …X r …X n , of different row lengths using x(k)
such that the row lengths are in the range 0.8 Hz to 2 Hz (the dominant frequency
PPG will be in this range)
(3) Perform SVD on each of these matrices X 1 , X 2 …X r …X n
(4) For each SVD, compute the singular value ratio (SVR) of the first two singular
values. Here SVR = .s 1 /s 2
(5) Plot the SVR against the row length r to obtain a graph called the SVR spectrum
of the signal.
(6) From the SVR spectrum, determine the value of row length, say r, for which the
SVR (s 1r /s 2r ) is maximum as seen in a typical SVR spectrum shown in Fig. 3.9.
(7) From the SVD of the corresponding data matrix X r reconstruct the PPG signal
by taking the average of all rows of s 1 u 1 v 1
t .
It should be noted here that the averaging process eliminates not only artefact
present in the PPG but also eliminates any additive noise in the PPG. Once one cycle
of a PPG is recovered, a kind of roll over procedure can be implemented by removing
the first r samples from and appending r new samples at the end.
For example, if the initial samples are x(k), k = 1…M then, after extracting the
first PPG cycle, x(k) is modified as: x(k), k = (r + 1) … (M + r). Once again steps 2