# Variable Fractional Delay

Delay input by time-varying fractional number of sample periods

• Library:
• DSP System Toolbox / Signal Operations

## Description

The Variable Fractional Delay block delays the input signal by a specified number of fractional samples along each channel of the input. The block can also concurrently compute multiple delayed versions (taps) of the same signal. For an example, see Delay Signal Using Multitap Fractional Delay.

When the delay has a fractional value, the block interpolates the input signal to obtain new samples at noninteger sampling intervals. You can set Interpolation mode parameter to one of `Linear`, `FIR`, or `Farrow`. The block supports time-varying delay values. That is, the delay value can vary within a frame from sample to sample.

The block assumes that the input values at the Delay port are between Dmin and Dmax, where Dmin appears in the ```Valid delay range``` section on the Main tab of the block dialog, and Dmax is the value of the Maximum delay (Dmax) in samples parameter. The block clips delay values less than Dmin to Dmin and delay values greater than Dmax to Dmax.

You must consider additional factors when selecting valid `Delay` values for the `FIR` and `Farrow` interpolation modes. For details, see Algorithms.

## Ports

### Input

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Specify the data input as a vector or matrix. The data input must have the same data type as the delay input.

This block supports variable-size input signal. That is, you can change the number of input rows during the simulation. However, the number of channels must remain constant.

Example: [1 2 3 4;5 1 4 2;2 6 2 3;1 2 3 2;3 4 5 6;1 2 3 1]

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fixed point`
Complex Number Support: Yes

Specify the delay input as a scalar, vector, matrix, or N-D array. The delay can be an integer or a fractional value. The block interpolates the signal to obtain new samples at noninteger sampling intervals. The delay input must have the same data type as the data input.

This block supports variable-size delay signal. That is, you can change one or both of the dimensions of the delay signal during simulation. However, the block must make sure that the resulting number of output channels remains constant throughout the simulation.

When the Input processing parameter is set to `Columns as channels (frame based)`, the table below shows the effect of the dimension of the delay input on the data input. For an example, see Delay Signal Using Multitap Fractional Delay.

Data InputDelay InputOutputEffect of Delay Input on Data Input
N (unoriented, one channel)scalarUnoriented (N)One delay value applied to the input channel
N (unoriented, one channel)Unoriented (N)Unoriented (N)Delay value varies within the frame from sample to sample
N (unoriented, one channel)1-by-PN-by-PP taps. Each column in the output is a delayed version of the input. The delay value is specified by the corresponding element in the delay input vector.
N (unoriented, one channel)N-by-PN-by-PP taps. In addition, delay varies within each frame from sample to sample.
N-by-1 (one channel with frame size equal to N)scalarN-by-1One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)Unoriented (N)N-by-1Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)N-by-1N-by-1Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)1-by-PN-by-PP taps. Each column in the output is a delayed version of the input. The delay value is specified by the corresponding element in the delay input vector.
N-by-1 (one channel with frame size equal to N)N-by-PN-by-PP taps. In addition, delay varies within each frame from sample to sample.
N-by-L (L channels with frame size equal to N)scalarN-by-LOne delay value applied to all input channels
N-by-L (L channels with frame size equal to N)1-by-LN-by-LUnique delay value for each input channel
N-by-L (L channels with frame size equal to N)N-by-1N-by-LDelay value varies within the frame from sample to sample. Same set of delay values for all channels.
N-by-L (L channels with frame size equal to N)N-by-LN-by-LDelay value varies within the frame from sample to sample. Different delay values for each input channel.
N-by-L (L channels with frame size equal to N)1-by-1-by-PN-by-L-by-PL channels. P taps per channel. Same delay for all channels.
N-by-L (L channels with frame size equal to N)1-by-L-by-PN-by-L-by-PL channels. P taps per channel. Taps vary across channels.
N-by-L (L channels with frame size equal to N)N-by-1-by-PN-by-L-by-PL channels. P taps per channel. Delay varies within the frame from sample to sample. Same set of delay values for each channel.
N-by-L (L channels with frame size equal to N)N-by-L-by-PN-by-L-by-PL channels. P taps per channel. Delay varies within the frame from sample to sample. Different set of delay values for each channel.

When the Input processing parameter is set to `Elements as channels (sample based)`, the table below shows the effect of the dimension of the delay input on the data input.

Data InputDelay InputOutputEffect of Delay Input on Data Input
N (unoriented, one channel)scalarUnoriented (N)One delay value applied to the input channel
N (unoriented, one channel)Unoriented (N)Unoriented (N)Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)scalarN-by-1One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)Unoriented (N)N-by-1Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)N-by-1N-by-1Delay value varies within the frame from sample to sample
N-by-L (L channels with N samples in each channel)scalarN-by-LOne delay value applied to all input channels
N-by-L (L channels with N samples in each channel)1-by-LN-by-LUnique delay value for each input channel
N-by-L (L channels with N samples in each channel)N-by-1N-by-LDelay value varies within the frame from sample to sample. Same set of delay values for all channels.
N-by-L (L channels with N samples in each channel)N-by-LN-by-LDelay value varies within the frame from sample to sample. Different delay values for each input channel.

Example: [2 3 4 5]

Example: [2.5]

Example: [5.6]

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fixed point`

### Output

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Delayed output, returned as a vector or matrix. The data type and complexity of the output match the data type and complexity of the data input.

When the Input processing parameter is set to `Columns as channels (frame based)`, the table below shows the effect of the dimension of the delay input on the data input.

Data InputDelay InputOutputEffect of Delay Input on Data Input
N (unoriented, one channel)scalarUnoriented (N)One delay value applied to the input channel
N (unoriented, one channel)Unoriented (N)Unoriented (N)Delay value varies within the frame from sample to sample
N (unoriented, one channel)1-by-PN-by-PP taps. Each column in the output is a delayed version of the input. The delay value is specified by the corresponding element in the delay input vector.
N (unoriented, one channel)N-by-PN-by-PP taps. In addition, delay varies within each frame from sample to sample.
N-by-1 (one channel with frame size equal to N)scalarN-by-1One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)Unoriented (N)N-by-1Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)N-by-1N-by-1Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)1-by-PN-by-PP taps. Each column in the output is a delayed version of the input. The delay value is specified by the corresponding element in the delay input vector.
N-by-1 (one channel with frame size equal to N)N-by-PN-by-PP taps. In addition, delay varies within each frame from sample to sample.
N-by-L (L channels with frame size equal to N)scalarN-by-LOne delay value applied to all input channels
N-by-L (L channels with frame size equal to N)1-by-LN-by-LUnique delay value for each input channel
N-by-L (L channels with frame size equal to N)N-by-1N-by-LDelay value varies within the frame from sample to sample. Same set of delay values for all channels.
N-by-L (L channels with frame size equal to N)N-by-LN-by-LDelay value varies within the frame from sample to sample. Different delay values for each input channel.
N-by-L (L channels with frame size equal to N)1-by-1-by-PN-by-L-by-PL channels. P taps per channel. Same delay for all channels.
N-by-L (L channels with frame size equal to N)1-by-L-by-PN-by-L-by-PL channels. P taps per channel. Taps vary across channels.
N-by-L (L channels with frame size equal to N)N-by-1-by-PN-by-L-by-PL channels. P taps per channel. Delay varies within the frame from sample to sample. Same set of delay values for each channel.
N-by-L (L channels with frame size equal to N)N-by-L-by-PN-by-L-by-PL channels. P taps per channel. Delay varies within the frame from sample to sample. Different set of delay values for each channel.

When the Input processing parameter is set to `Elements as channels (sample based)`, the table below shows the effect of the dimension of the delay input on the data input.

Data InputDelay InputOutputEffect of Delay Input on Data Input
N (unoriented, one channel)scalarUnoriented (N)One delay value applied to the input channel
N (unoriented, one channel)Unoriented (N)Unoriented (N)Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)scalarN-by-1One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)Unoriented (N)N-by-1Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)N-by-1N-by-1Delay value varies within the frame from sample to sample
N-by-L (L channels with N samples in each channel)scalarN-by-LOne delay value applied to all input channels
N-by-L (L channels with N samples in each channel)1-by-LN-by-LUnique delay value for each input channel
N-by-L (L channels with N samples in each channel)N-by-1N-by-LDelay value varies within the frame from sample to sample. Same set of delay values for all channels.
N-by-L (L channels with N samples in each channel)N-by-LN-by-LDelay value varies within the frame from sample to sample. Different delay values for each input channel.

Example: [0 0 0 0;0 0 0 0;1 0 0 0;5 2 0 0;2 1 3 0;1 6 4 4]

Example: [0 0 0 0;0 0 0 0;0.5 1.0 1.5 2.0;3 1.5 3.5 3.0;3.5 3.5 3.0 2.5;1.5 4.0 2.5 2.5]

Example: [0 0 0 0;0 0 0 0;0 0 0 0;0 0 0 0;0 0 0 0;0.4 0.8 1.2 1.6]

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fixed point`
Complex Number Support: Yes

## Parameters

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Specify the method of interpolation. Using this method, the block interpolates the signal to obtain new samples at noninteger sampling intervals.

• `Linear` –– Linear interpolation. In this mode, the block stores the Dmax+1 most recent samples the In port receives for each channel. Dmax is the value you specify in the Maximum delay (Dmax) in samples parameter.

• `FIR` –– Polyphase FIR interpolation. In this mode, the block stores the Dmax+P+1 most recent samples the In port receives for each channel. P is the value you specify in the Interpolation filter half-length (P) parameter.

• `Farrow` –– LaGrange method. In this mode, the block stores the Dmax+$\frac{N}{2}$+1 most recent samples the In port receives for each channel. N is the value you specify in the Farrow filter length (N) parameter.

For more details on these methods, see Algorithms.

Half-length of the FIR interpolation filter. For periodic signals, a larger value of this property, which indicates a higher order filter, yields a better estimate of the delayed output sample. A property value of 4 to 6, which corresponds to a 7th-order to 11th-order filter, is usually adequate.

Example: `6`

Example: `10`

#### Dependencies

This parameter applies only when you set Interpolation mode to `FIR`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `Boolean` | `fixed point`

Number of interpolation points per input sample at which a unique FIR interpolation filter is computed.

Example: `20`

Example: `5`

#### Dependencies

This parameter applies only when you set Interpolation mode to `FIR`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`

Normalized input bandwidth at which to constrain the interpolated output samples. A value of `1` equals the Nyquist frequency, or half the sampling frequency, Fs. Use this property to take advantage of the bandlimited frequency content of the input. For example, if the input signal does not have frequency content above Fs/4, you can specify a value of `0.5`.

Example: `0.5`

Example: `0.8`

#### Dependencies

This parameter applies only when you set Interpolation mode to `FIR`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `Boolean` | `fixed point`

Length of the FIR filter implemented using the Farrow structure. If the length equals 2, the filter performs linear interpolation.

Example: `4`

Example: `10`

#### Dependencies

This parameter applies only when you set Interpolation mode to `Farrow`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32`

Maximum delay the block can produce, Dmax. Input delay values exceeding this maximum are clipped to Dmax.

Example: 200

Example: 500

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32`

Specify how the block should process the input. You can set this parameter to one of the following options:

• `Columns as channels (frame based)` (default) — When you select this option, the block treats each column of the input as a separate channel. The block treats each of the R input columns as independent channels containing Mi sequential time samples.

The input to the Delay port, v, contains floating-point values that specify the number of sample intervals to delay the current input.

The input to the Delay port can be a scalar value to uniformly delay every sample in every channel. It can also be a length-M column vector, containing one delay for each sample in the input frame. The block applies the set of delays contained in the vector identically to every channel of a multichannel input. The Delay port entry can also be a length-R row vector, containing one delay for each channel. Finally, the Delay port entry can be an M-by-R matrix, containing a different delay for each corresponding element of the input.

For example, if v is the Mi-by-1 matrix `[v(1) v(2) ... v(Mi)]'`, the earliest sample in the current frame is delayed by `v(1)` fractional sample intervals, the following sample in the frame is delayed by `v(2)` fractional sample intervals, and so on. The block applies the set of fractional delays contained in v identically to every channel of a multichannel input.

• `Elements as channels (sample based)` –– When you select this option, the block treats each element of the input as a separate channel. The block treats each element of the N-D input array, u, as an independent channel. The input to the Delay port, v, must either be an N-D array of the same size and dimension as the input u, or be a scalar value, such that DminvDmax.

For example, consider an M-by-R input matrix. The block treats each of the M*R matrix elements as independent channels. The input to the Delay port can be an M-by-R matrix of floating-point values in the range DminvDmax that specifies the number of sample intervals to delay each channel of the input, or it can be a scalar floating-point value, DminvDmax, by which to equally delay all channels.

In sample-based processing mode, the block treats an unoriented vector input as an M-by-1 matrix. In this mode, the output is also an unoriented vector.

Specify the values with in the block's memory at the start of the simulation. The dimensions of this parameter can vary depending on whether you want fixed or time-varying initial conditions. The block treats each of the R input columns as a frame containing M sequential time samples from an independent channel.

For an M-by-R input matrix, u, you can set this parameter as follows:

• To specify fixed initial conditions, set this parameter to a scalar value. The block initializes every sample of every channel in memory using the value you specify.

• The dimensions you specify for time-varying initial conditions depend on the interpolation method. To specify different time-varying initial conditions for each channel, set this parameter as follows:

• If you set the Interpolation mode to `Linear`, set the Initial conditions to an array of size `1`-by-R-by-D, where D is the value in Maximum delay (Dmax) in samples parameter.

• If you set the Interpolation mode to `FIR` or `Farrow`, set the Initial conditions to an array of size 1-by-R-by-(D+L), where D is the value of the maximum delay. For FIR interpolation, `L` is the value of the interpolation filter half length. For Farrow interpolation, `L` equals `floor` of half the value of the farrow filter length (```floor( farrow filter length/2)```).

Example: `1`

Example: randn(1,3,104)

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32`

Select this box to disable direct feedthrough by increasing the minimum possible delay value. When you set the Input processing parameter to `Columns as channels (frame based)`, the block increases the minimum possible delay value by ```frame size```  – 1. Similarly, when you set the Input processing parameter to ```Elements as channels (sample based)```, the block increases the minimum possible delay value by one sample.

Checking this box allows you to use the Variable Fractional Delay block in feedback loops.

Specify the block's behavior when the input delay values are too small to center the kernel.

You can specify how the block handles input delay values that are too small for the kernel to be centered using one of the following choices:

• In both `FIR` and `Farrow` interpolation modes, you can select ```Clip to the minimum value necessary for centered kernel```. This option forces the block to increase Dmin to the smallest value necessary to keep the kernel centered.

• In `FIR` interpolation mode, you can select ```Switch to linear interpolation if kernel cannot be centered```. This option forces the block to preserve the value of Dmin and compute all interpolated values using `Linear` interpolation.

• In `Farrow` interpolation mode, you can select `Use off-centered kernel`. This option forces the block to preserve the value of Dmin and compute the interpolated values using a farrow filter with an off-centered kernel.

#### Dependencies

This parameter applies only when Interpolation mode is set to `FIR` or `Farrow`.

The delay range values [Dmin Dmax] are calculated (in samples) by the block based on the current parameter settings. Dmin is the smallest possible valid delay value (in samples). The block clips all input delay values less than Dmin to Dmin. Dmax is the maximum valid delay value (in samples). The block clips all input delay values greater than Dmax to Dmax.

When the Interpolation mode is set to one of the following:

• `Linear` –– Dmin equal 0. Dmax equals the value you specify in the Maximum delay (Dmax) in samples parameter.

• `FIR` –– Dmin equals P – 1, where P is the value you specify in Interpolation filter half-length (P). Dmax equals the value you specify in the Maximum delay (Dmax) in samples parameter.

• `Farrow` –– Dmin equals N/2 – 1, where N is the value you specify in Farrow filter length (N). Dmax equals the value you specify in the Maximum delay (Dmax) in samples parameter.

Example: [1 100]

Example: [2 100]

Example: [3 100]

### Fixed-Point Properties

Fixed-Point Properties

Specify the rounding mode for fixed-point operations as one of the following:

• `Zero`

• `Ceiling`

• `Convergent`

• `Floor`

• `Nearest`

• `Round`

• `Simplest`

For more details, see rounding mode.

When you select this parameter, the block saturates the result of its fixed-point operation. When you clear this parameter, the block wraps the result of its fixed-point operation. For details on `saturate` and `wrap`, see overflow mode for fixed-point operations.

Specify the data type of the filter coefficients as one of the following:

• `Same word length as input` –– The word length of the filter coefficients matches that of the input to the block. The fraction length of the coefficients is automatically set to the binary-point only scaling that provides you with the best precision possible given the value and word length of the coefficients.

• `Specify word length` –– Specify the word length of the coefficients, in bits. In this mode, the fraction length of the coefficients is automatically set to the binary-point only scaling that provides you with the best precision possible given the value and word length of the coefficients.

For more information on the coefficients data type this block uses, see the Fixed Point section.

Specify the data type of the product output as one of the following:

• `Same as first input` –– The block specifies the product output data type to be the same as that of the data input.

• `Binary point scaling` –– Specify the word length and the fraction length of the product output, in bits.

For more information on the product output data type, see Multiplication Data Types and the Fixed Point section.

Specify the data type of an accumulation operation as one of the following:

• `Same as product output` –– The block specifies the accumulator data type to be the same as that of the product output data type.

• `Same as first input` –– The block specifies the accumulator data type to be the same as that of the data input.

• `Binary point scaling` –– Specify the word length and the fraction length of the accumulator output, in bits.

For more information on the accumulator data type this block uses, see the Fixed Point.

Specify the data type of the product polynomial value as one of the following:

• `Same as first input` –– The block specifies the product polynomial value data type to be the same as that of the data input.

• `Binary point scaling` –– Specify the word length and the fraction length of the product output polynomial, in bits.

For more information on the product polynomial value data type this block uses, see the Fixed Point section.

#### Dependencies

This property applies when you set Interpolation mode to `Farrow`.

Specify the data type of the accumulator polynomial value as one of the following:

• `Same as first input` –– The block specifies the accumulator polynomial value data type to be the same as that of the data input.

• `Binary point scaling` –– Specify the word length and the fraction length of the accumulator polynomial value, in bits.

For more information on the accumulator polynomial value data type that this block uses, see the Fixed Point section.

#### Dependencies

This property applies when you set Interpolation mode to `Farrow`.

Specify the data type of the multiplicand polynomial value as one of the following:

• `Same as first input` –– The block specifies the multiplicand polynomial value data type to be the same as that of the data input.

• `Binary point scaling` –– Specify the word length and the fraction length of the multiplicand polynomial value, in bits.

For more information on the multiplicand polynomial value data type this block uses, see the Fixed Point section.

#### Dependencies

This property applies when you set Interpolation mode to `Farrow`.

Specify the data type of the block output as one of the following:

• `Same as accumulator` –– The block specifies the output data type to be the same as that of the accumulator output data type.

• `Same as first input` –– The block specifies the output data type to be the same as that of the data input.

• `Binary point scaling` –– Specify the word length and the fraction length of the block output, in bits.

For more information on the output data type this block uses, see the Fixed Point section.

Select this parameter to prevent the fixed-point tools from overriding the data types you specify in the block dialog box.

## Block Characteristics

 Data Types `double` | `fixed point` | `integer` | `single` Direct Feedthrough `no` Multidimensional Signals `no` Variable-Size Signals `yes` Zero-Crossing Detection `no`

## Algorithms

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The delay value specified at the Delay port serves as an index into the block's memory, U, which stores, at a minimum, the Dmax+1 most recent samples received at the In port for each channel. For example, an integer delay of 5 on a scalar input sequence retrieves and outputs the fifth most recent input sample from the block's memory, U(6). The block computes fractional delays by interpolating between stored samples. The block uses a linear, FIR, or farrow interpolation method to interpolate signal values at noninteger sample intervals.

## Extended Capabilities

### Blocks

Introduced before R2006a

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