# dsp.Differentiator

Direct form FIR fullband differentiator filter

## Description

The `dsp.Differentiator`

System object™ applies a fullband differentiator filter on the input signal to differentiate
all its frequency components. This object uses an FIR equiripple filter design to design the
differentiator filter. The ideal frequency response of the differentiator is $$D(\omega )=j\omega $$ for $$-\pi \le \omega \le \pi $$. You can design the filter with minimum order with a specified order. This
object supports fixed-point operations.

To filter each channel of your input:

Create the

`dsp.Differentiator`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Description

returns a
differentiator, `DF`

= dsp.Differentiator`DF`

, which independently filters each channel of the
input over time using the given design specifications.

sets each property name to the specified value. Unspecified properties have default
values.`DF`

= dsp.Differentiator(`Name,Value`

)

## Properties

Unless otherwise indicated, properties are *nontunable*, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
`release`

function unlocks them.

If a property is *tunable*, you can change its value at
any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`DesignForMinimumOrder`

— Design minimum order filter

`true`

(default) | `false`

Option to design a minimum-order filter, specified as a logical scalar. The filter has 2 degrees of freedom. When you set this property to

`true`

— The object designs the filter with the minimum order that meets the`PassbandRipple`

value.`false`

— The object designs the filter with order that you specify in the`FilterOrder`

property.

This property is not tunable.

`FilterOrder`

— Order of the filter

31 (default) | odd positive integer

Order of the filter, specified as an odd positive integer.

This property is not tunable.

#### Dependencies

You can specify the filter order only when
`'DesignForMinimumOrder'`

is set to
`false`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`PassbandRipple`

— Maximum passband ripple

0.1 (default) | positive real scalar

Maximum passband ripple in dB, specified as a positive real scalar.

This property is not tunable.

#### Dependencies

You can specify the passband ripple only when
`'DesignForMinimumOrder'`

is set to `true`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`ScaleCoefficients`

— Scale filter coefficients

`false`

(default) | `true`

Option to scale the filter coefficients, specified as a logical scalar. When you set
this property to `true`

, the object scales the filter coefficients to
preserve the input dynamic range.

This property is not tunable.

### Fixed-Point Properties

`CoefficientsDataType`

— Word and fraction lengths of coefficients

`numerictype(1,16)`

(default) | `numerictype`

object

Word and fraction lengths of coefficients, specified as a signed or unsigned
`numerictype`

object. The default,
`numerictype(1,16)`

, corresponds to a signed numeric type object
with 16-bit coefficients. To give the best possible precision, the fraction length is
computed based on the coefficient values.

This property is not tunable.

The word length of the output is the same as the word length of the input. The object computes the fraction length of the output such that the entire dynamic range of the output can be represented without overflow. For details on how the object computes the fraction length of the output, see Fixed-Point Precision Rules for Avoiding Overflow in FIR Filters.

`RoundingMethod`

— Rounding method for output fixed-point operations

`'Floor'`

(default) | `'Ceiling'`

| `'Convergent'`

| `'Nearest'`

| `'Round'`

| `'Simplest'`

| `'Zero'`

Rounding method for output fixed-point operations, specified as a character vector. For more information on the rounding modes, see Precision and Range.

This property is not tunable.

## Usage

### Syntax

### Description

### Input Arguments

`x`

— Data input

vector | matrix

Data input, specified as a vector or a matrix. If the input signal is a matrix, each column of the matrix is treated as an independent channel. The number of rows in the input signal denotes the channel length. The data type characteristics (double, single, or fixed-point) and the real-complex characteristics (real or complex valued) must be the same for the input data and output data.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `fi`

**Complex Number Support: **Yes

### Output Arguments

`y`

— Differentiated signal

vector | matrix

Differentiated signal, returned as a vector or matrix of the same size, data type,
and complexity as the input signal, `x`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `fi`

**Complex Number Support: **Yes

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

## Examples

### Group Delay Estimation

**Note**: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent `step`

syntax. For example, myObject(x) becomes step(myObject,x).

Estimate the group delay of a linear phase FIR filter using a `dsp.TransferFunctionEstimator`

object followed by `dsp.PhaseExtractor`

and `dsp.Differentiator`

objects. The group delay of a linear phase FIR filter is given by , where is the phase information of the filter, is the frequency vector, and *N* is the order of the filter.

**Set Up the Objects**

Create a linear phase FIR lowpass filter. Set the order to 200, the passband frequency to 255 Hz, the passband ripple to 0.1 dB, and the stopband attenuation to 80 dB. Specify a sample rate of 512 Hz.

Fs = 512; LPF = dsp.LowpassFilter('SampleRate',Fs,'PassbandFrequency',255,... 'DesignForMinimumOrder',false,'FilterOrder',200);

To estimate the transfer function of the lowpass filter, create a transfer function estimator. Specify the window to be `Hann`

. Set the FFT length to 1024 and the number of spectral averages to 200.

TFE = dsp.TransferFunctionEstimator('FrequencyRange','twosided',... 'SpectralAverages',200,'FFTLengthSource','Property',... 'FFTLength',1024);

To extract the unwrapped phase from the frequency response of the filter, create a phase extractor.

PE = dsp.PhaseExtractor;

To differentiate the phase , create a differentiator filter. This value is used in computing the group delay.

DF = dsp.Differentiator;

To smoothen the input, create a variable bandwidth FIR filter.

Gain1 = 512/pi; Gain2 = -1; VBFilter = dsp.VariableBandwidthFIRFilter('CutoffFrequency',10,... 'SampleRate',Fs);

To view the group delay of the filter, create an array plot object.

AP = dsp.ArrayPlot('PlotType','Line','YLimits',[-500 400],... 'YLabel','Amplitude','XLabel','Number of samples');

**Run the Algorithm**

The `for`

-loop is the streaming loop that estimates the group delay of the filter. In the loop, the algorithm filters the input signal, estimates the transfer function of the filter, and differentiates the phase of the filter to compute the group delay.

Niter = 1000; % Number of iterations for k = 1:Niter x = randn(512,1); % Input signal = white Gaussian noise y = LPF(x); % Filter noise with Lowpass FIR filter H = TFE(x,y); % Compute transfer function estimate Phase = PE(H); % Extract the Unwrapped phase phaseaftergain1 = Gain1*Phase; DiffOut = DF(phaseaftergain1); % Differentiate the phase phaseaftergain2 = Gain2 * DiffOut; VBFOut = VBFilter(phaseaftergain2); % Smooth the group delay AP(VBFOut); % Display the group delay end

As you can see, the group delay of the lowpass filter is 100.

### Convert FM Signal to AM Signal

Create an FM wave on a 100 Hz carrier signal sampled at 1.5 kHz.

Fc = 1e2; % Carrier Fs = 1.5e3; % Sample rate sinewave = dsp.SineWave('Frequency',10,... 'SamplesPerFrame',1e3,... 'SampleRate',Fs);

Convert the FM signal to an AM signal.

ts = timescope('TimeSpanSource','Property',... 'TimeSpan',0.3,... 'BufferLength',10*Fs,... 'SampleRate',Fs,... 'ShowGrid',true,... 'YLimits',[-1.5 1.5],... 'LayoutDimensions',[2 1]); df = dsp.Differentiator; tic while toc<2.2 x = step(sinewave); fm_y = modulate(x,Fc,Fs,'fm'); am_y = step(df,fm_y); step(ts,fm_y,am_y); end release(df); release(ts);

## Algorithms

### Differentiator Filter

Differentiator computes the derivative of a signal. The frequency response of an ideal differentiator filter is given by $$D(\omega )=j\omega $$, defined over the Nyquist interval $$-\pi \le \omega \le \pi $$.

The frequency response is antisymmetric and is linearly proportional to the frequency.

`dsp.Differentiator`

object acts as a differentiator
filter. This object condenses the two-step process into one. For the
minimum order design, the object uses generalized Remez FIR filter
design algorithm. For the specified order design, the object uses
the Parks-McClellan optimal equiripple FIR filter design algorithm.
The filter is designed as a linear phase Type-IV FIR filter with a
Direct form structure.

The ideal differentiator has an antisymmetric impulse response given by $$d(n)=-d(-n)$$. Hence $$d(0)=0$$. The differentiator must have zero response at zero frequency.

**Linear-Phase FIR Differentiator Filter**

The impulse response of an antisymmetric linear-phase FIR filter
is given by $$h(n)=-h(M-1-n)$$, where *M* is
the length of the filter. Because the filter is antisymmetric, you
can use this type of FIR filter to design the linear-phase FIR differentiators.

Consider the design of linear-phase FIR differentiators based on the Chebyshev approximation criterion.

If *M* is odd, the real-valued frequency response
of the FIR filter, H_{r}(ω),
has the characteristics that H_{r}(0)
= 0 and H_{r}(π)
= 0. This filter satisfies the condition of zero
response at zero frequency. However, it is not fullband because H_{r}(π)
= 0. This differentiator has a linear response
over the limited frequency range [0 2π*f _{p}*],
where

*f*is the bandwidth of the differentiator. The absolute error between the desired response and the Chebyshev approximation increases as

_{p}*ω*increases from 0 to 2π

*f*.

_{p}If *M* is even, the real-valued frequency response
of the FIR filter, H_{r}(ω),
has the characteristics that H_{r}(0)
= 0 and H_{r}(π)
≠ 0. This filter satisfies the condition
of zero response at zero frequency. It is fullband and this design
results in a significantly smaller approximation error than comparable
odd-length differentiators. Hence, even-length (odd order) differentiators
are preferred in practical systems.

## References

[1] Orfanidis, Sophocles J.* Introduction to Signal Processing*.
Upper Saddle River, NJ: Prentice-Hall, 1996.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).

This object also supports SIMD code generation using Intel AVX2 technology when the
input signal has a data type of `single`

or
`double`

.

The SIMD technology significantly improves the performance of the generated code.

### Fixed-Point Conversion

Design and simulate fixed-point systems using Fixed-Point Designer™.

## See Also

### Functions

### Objects

`dsp.HighpassFilter`

|`dsp.VariableBandwidthFIRFilter`

|`dsp.VariableBandwidthIIRFilter`

|`dsp.FIRFilter`

|`dsp.BiquadFilter`

### Blocks

**Introduced in R2016a**

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