deconv
Least-squares deconvolution and polynomial division
Description
Polynomial Long Division
[
deconvolves a
vector x
,r
] =
deconv(y
,h
)h
out of a vector y
using
polynomial long division, and returns the quotient x
and
remainder r
such that y = conv(x,h) + r
.
If y
and h
are vectors of polynomial
coefficients, then deconvolving them is equivalent to dividing the polynomial
represented by y
by the polynomial represented by
h
.
Least-Squares Deconvolution
Since R2023b
[
specifies the subsections of the convolved signal x
,r
] =
deconv(y
,h
,shape
)y
, where
y = conv(x,h,shape) + r
.
If you use the least-squares deconvolution method
(Method="least-squares"
), then you can specify
shape
as "full"
,
"same"
, or "valid"
. Otherwise, if you
use the default long-division deconvolution method
(Method="long-division"
), then shape
must be "full"
.
[
specifies options
using one or more name-value arguments in addition to any of the input argument
combinations in previous syntaxes.x
,r
] =
deconv(___,Name=Value
)
You can specify the deconvolution method using
deconv(__,Method=algorithm)
, wherealgorithm
can be"long-division"
or"least-squares"
.You can also specify the Tikhonov regularization factor to the least-squares solution of the deconvolution method using
deconv(__,RegularizationFactor=alpha)
.
Examples
Polynomial Division
Create two vectors, y
and h
, containing the coefficients of the polynomials and , respectively. Divide the first polynomial by the second by deconvolving h
out of y
. The deconvolution results in quotient coefficients corresponding to the polynomial and remainder coefficients corresponding to .
y = [2 7 4 9]; h = [1 0 1]; [x,r] = deconv(y,h)
x = 1×2
2 7
r = 1×4
0 0 2 2
Least-Squares Deconvolution of Fully Convolved Signal
Since R2023b
Create a signal x
that has a Gaussian shape. Convolve this signal with an impulse response h
that consists of random noise.
N = 200;
n = 0.1*(1:N);
rng("default")
x = 2*exp(-0.5*((n-10)).^2);
h = 0.1*randn(1,length(x));
y = conv(x,h);
Plot the original signal, the impulse response, and the convolved signal.
figure tiledlayout(3,1) nexttile plot(n,x) title("Original Signal") nexttile plot(n,h) title("Impulse Response") nexttile plot(0.1*(1:length(y)),y) title("Convolved Signal")
Next, find the deconvolution of signal y
with respect to impulse response h
using the default polynomial long-division method. Using this method, the deconvolution computation is unstable, and the result can rapidly increase.
[x1,r1] = deconv(y,h); x1(end)
ans = 7.5992e+90
Instead, find the deconvolution using the least-squares method for a numerically stable computation.
[x2,r2] = deconv(y,h,Method="least-squares");
Plot both deconvolution results. Here, the least-squares method correctly returns the original signal that has a Gaussian shape.
figure tiledlayout(2,1) nexttile plot(n,x1) title("Deconvolved Signal Using ""long-division"" Method") nexttile plot(n,x2) title("Deconvolved Signal Using ""least-squares"" Method")
Least-Squares Deconvolution of Central Part of Convolved Signal
Since R2023b
Create two vectors. Find the central part of the convolution of xin
and h
that is the same size as xin
. The central part of the convolved signal y has a length of 7 instead of the full length, which is length(xin)+length(h)-1
, or 10.
xin = [-1 2 3 -2 0 1 2];
h = [2 4 -1 1];
y = conv(xin,h,"same")
y = 1×7
15 5 -9 7 6 7 -1
Find the least-squares deconvolution of signal y
with respect to impulse response h
. Use the "same"
option to specify that the convolved signal y
is only the central part, where y = conv(x,h,"same") + r
. Show that deconv
recovers the original signal in x
within round-off error.
[x,r] = deconv(y,h,"same",Method="least-squares")
x = 1×7
-1.0000 2.0000 3.0000 -2.0000 0.0000 1.0000 2.0000
r = 1×7
10-14 ×
0 0.0888 0.1776 0 0 0 0
Least-Squares Deconvolution Problem with Infinite Solutions
Since R2023b
Create two vectors, each with two elements, and convolve them using the "valid"
option. This option returns only those parts of the convolution that are computed without the zero-padded edges. In this case, the convolved signal y
has only one element.
xin = [-1 2];
h = [2 5];
y = conv(xin,h,"valid")
y = -1
Find the least-squares deconvolution of convolved signal y
with respect to impulse response h
. With the "valid"
option, deconv
does not always return the original signal in x
, but it returns the solution of the deconvolution problem that minimizes norm(x)
instead.
[x,r] = deconv(y,h,"valid",Method="least-squares")
x = 1×2
-0.1724 -0.0690
r = -3.3307e-16
To check the solution, you can find the full convolution of the computed signal x
with h
. The central part of this convolved signal is the same as the original y
that defined the deconvolution problem.
yfull = conv(x,h,"full")
yfull = 1×3
-0.3448 -1.0000 -0.3448
In this problem, deconv
returns a different signal than the original signal because it solves for one equation with two variables, which is . This system is underdetermined, meaning this system has more variables than equations. This system has infinite solutions when using the least-squares method to minimize the residual norm, or norm(y - conv(x,h,"valid"))
, to 0. For this reason, deconv
also finds a solution that minimizes norm(x)
.
The following figure illustrates the situation for this underdetermined problem. The blue line represents the infinite number of solutions to the equation . The orange circle represents the minimum distance from the origin to the line of solutions. The solution returned by deconv
lies at the tangent point between the line and circle, indicating the solution closest to the origin.
Specify Regularization Factor for Noisy Signal
Since R2023b
Create two signals, x
and h
, and convolve them. Add some random noise to the convolved signal in y
.
N = 200;
n = 0.1*(1:N);
rng("default")
x = 2*exp(-0.8*(n - 8).^2) - 4*exp(-2*(n - 10).^2);
h = 2.*exp(-1*(n - 5).^2).*cos(4*n);
y = conv(x,h);
y = y + max(y)*0.05*randn(1,length(y));
Plot the original signal, the impulse response, and the convolved signal.
figure tiledlayout(3,1) nexttile plot(n,x) title("Original Signal") nexttile plot(n,h) title("Impulse Response") nexttile plot(0.1*(1:length(y)),y) title("Convolved Signal with Added Noise")
Next, find the deconvolution of the noisy signal y
with respect to the impulse response h
by using the least-squares method without a regularization factor. By default, the regularization factor is 0.
[x1,r1] = deconv(y,h,Method="least-squares");
Plot the original signal and the deconvolved signal. Here, the deconv
function without a regularization factor cannot recover the original signal from the noisy signal.
figure; tiledlayout(3,1); nexttile plot(n,x) title("Original Signal") nexttile plot(n,x1) title("Deconvolved Signal Without Regularization");
Instead, find the deconvolution of y
with respect to h
by using the least-squares method with a regularization factor of 1
. For an ill-conditioned deconvolution problem, such as one that involves noisy signal, you can specify a regularization factor so that overfitting does not occur in the least-squares solution.
[x2,r2] = deconv(y,h,Method="least-squares",RegularizationFactor=1);
Plot this deconvolved signal. Here, the deconv
function with a specified regularization factor recovers the original signal.
nexttile
plot(n,x2)
title("Deconvolved Signal Using Regularization")
Input Arguments
y
— Input signal to be deconvolved
row or column vector
Input signal to be deconvolved, specified as a row or column vector.
Data Types: double
| single
Complex Number Support: Yes
h
— Impulse response or filter used for deconvolution
row or column vector
Impulse response or filter used for deconvolution, specified as a row or
column vector. y
and h
can have
different lengths and data types.
If one or both of
y
andh
are of typesingle
, then the outputs are also of typesingle
. Otherwise, the outputs are of typedouble
.The lengths of the inputs should satisfy
length(h) <= length(y)
. However, iflength(h) > length(y)
, thendeconv
returns the outputs asx = 0
andr = y
.
Data Types: double
| single
Complex Number Support: Yes
shape
— Subsection of convolved signal
"full"
(default) | "same"
| "valid"
Since R2023b
Subsection of the convolved signal, specified as one of these values:
"full"
(default) —y
contains the full convolution ofx
withh
."same"
—y
contains the central part of the convolution that is the same size asx
."valid"
—y
contains only those parts of the convolution that are computed without the zero-padded edges. Using this option,length(y)
ismax(length(x)-length(h)+1,0)
, except whenlength(h)
is zero. Iflength(h) = 0
, thenlength(y) = length(x)
.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Example: [x,r] =
deconv(y,h,Method="least-squares",RegularizationFactor=1e-3)
Method
— Deconvolution method
"long-division"
(default) | "least-squares"
Since R2023b
Deconvolution method, specified as one of these values:
"long-division"
— Deconvolution by polynomial long division (default)."least-squares"
— Deconvolution by least squares, where the deconvolved signalx
is computed to minimize the norm of the residual signal (or remainder)r
. That is,x
is the solution that minimizesnorm(y - conv(x,h))
.
RegularizationFactor
— Tikhonov regularization factor
0
(default) | real number
Since R2023b
Tikhonov regularization factor for least-squares deconvolution,
specified as a real number. When using the least-squares deconvolution
method, specifying the regularization factor as alpha
returns a vector x
that minimizes norm(r)^2
+ norm(alpha*x)^2
. For ill-conditioned problems,
specifying the regularization factor gives preference to solutions
x
with smaller norms.
If you use the default long-division deconvolution method, then
RegularizationFactor
must be
0
.
Data Types: double
| single
Output Arguments
x
— Deconvolved signal or quotient from division
row or column vector
Deconvolved signal or quotient from division, returned as a row or column
vector such that y = conv(x,h) + r
.
Data Types: double
| single
r
— Residual signal or remainder from division
row or column vector
Residual signal or remainder from division, returned as a row or column
vector such that y = conv(x,h) + r
.
Data Types: double
| single
References
[1] Nagy, James G. “Fast Inverse QR Factorization for Toeplitz Matrices.” SIAM Journal on Scientific Computing 14, no. 5 (September 1993): 1174–93. https://doi.org/10.1137/0914070.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The deconv
function
supports GPU array input with these usage notes and limitations:
The
"least-squares"
deconvolution method is not supported.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
The
"least-squares"
deconvolution method is not supported.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006aR2023b: Perform least-squares deconvolution with different convolved subsections
You can now perform least-squares deconvolution by specifying the
Method
name-value argument as
"least-squares"
. You can also specify different convolved
subsections and the Tikhonov regularization factor with least-squares
deconvolution.
In previous releases, deconv
can perform deconvolution using
only a polynomial long-division method. The new arguments allow you to perform
least-squares deconvolution (Method="least-squares"
), which
returns more stable solutions compared to the default long-division deconvolution
(Method="long-division"
).
When you use the least-squares method to deconvolve a signal y
with respect to an impulse response h
, deconv
returns the signal x
that minimizes the norm of the residual
signal (or remainder) r = y - conv(x,h)
. That is,
x
is the solution that minimizes norm(r)
.
You can also specify the Tikhonov regularization factor alpha
to
return a solution x
that minimizes norm(r)^2 +
norm(alpha*x)^2
for ill-conditioned problems.
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