# 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)`

, where`algorithm`

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 $2{\mathit{x}}^{3}+7{\mathit{x}}^{2}+4\mathit{x}+9$ and ${\mathit{x}}^{2}+1$, respectively. Divide the first polynomial by the second by deconvolving `h`

out of `y`

. The deconvolution results in quotient coefficients corresponding to the polynomial $$2x+7$$ and remainder coefficients corresponding to $$2x+2$$.

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 =1×710^{-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 $$-1=5\cdot x(1)+2\cdot x(2)$$. 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 $$x(2)=-1/2-5/2\cdot x(1)$$. 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`

and`h`

are of type`single`

, then the outputs are also of type`single`

. Otherwise, the outputs are of type`double`

.The lengths of the inputs should satisfy

`length(h) <= length(y)`

. However, if`length(h) > length(y)`

, then`deconv`

returns the outputs as`x = 0`

and`r = 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 of`x`

with`h`

.`"same"`

—`y`

contains the central part of the convolution that is the same size as`x`

.`"valid"`

—`y`

contains only those parts of the convolution that are computed without the zero-padded edges. Using this option,`length(y)`

is`max(length(x)-length(h)+1,0)`

, except when`length(h)`

is zero. If`length(h) = 0`

, then`length(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 signal`x`

is computed to minimize the norm of the residual signal (or remainder)`r`

. That is,`x`

is the solution that minimizes`norm(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™.

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 R2006a**

### R2023b: 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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