Variable Fractional Delay

Delay input by time-varying fractional number of sample periods

Libraries:
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 D_min and D_max, where D_min appears in the Valid delay range section on the Main tab of the block dialog, and D_max is the value of the Maximum delay (Dmax) in samples parameter. The block clips delay values less than D_min to D_min and delay values greater than D_max to D_max.

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

Examples

Delay Signal Using Multitap Fractional Delay

Concurrently delay an input signal using multiple taps.

Open Script

Ports

Input

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In — Data input
vector | matrix

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]

Delay — Delay input
scalar | vector | matrix | N-D array

Specify the delay input in samples 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.

The block dialog box displays the valid delay range in samples. The block determines this range based on the settings in the block dialog box. If you specify a delay value that is less than the minimum acceptable value, the block performs the action you choose in the For small input delay values parameter. If you set the Interpolation mode to Linear, the minimum acceptable value is 0.

If you specify a delay value that is greater than the maximum acceptable value, the block clips the delay value to the maximum value.

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 Input	Delay Input	Output	Effect of Delay Input on Data Input
N (unoriented, one channel)	scalar	Unoriented (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-P	N-by-P	P 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-P	N-by-P	P taps. In addition, delay varies within each frame from sample to sample.
N-by-1 (one channel with frame size equal to N)	scalar	N-by-1	One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)	Unoriented (N)	N-by-1	Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)	N-by-1	N-by-1	Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)	1-by-P	N-by-P	P 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-P	N-by-P	P taps. In addition, delay varies within each frame from sample to sample.
N-by-L (L channels with frame size equal to N)	scalar	N-by-L	One delay value applied to all input channels
N-by-L (L channels with frame size equal to N)	1-by-L	N-by-L	Unique delay value for each input channel
N-by-L (L channels with frame size equal to N)	N-by-1	N-by-L	Delay 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-L	N-by-L	Delay 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-P	N-by-L-by-P	L 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-P	N-by-L-by-P	L channels. P taps per channel. Taps vary across channels.
N-by-L (L channels with frame size equal to N)	N-by-1-by-P	N-by-L-by-P	L 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-P	N-by-L-by-P	L 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 Input	Delay Input	Output	Effect of Delay Input on Data Input
N (unoriented, one channel)	scalar	Unoriented (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)	scalar	N-by-1	One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)	Unoriented (N)	N-by-1	Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)	N-by-1	N-by-1	Delay value varies within the frame from sample to sample
N-by-L (L channels with N samples in each channel)	scalar	N-by-L	One delay value applied to all input channels
N-by-L (L channels with N samples in each channel)	1-by-L	N-by-L	Unique delay value for each input channel
N-by-L (L channels with N samples in each channel)	N-by-1	N-by-L	Delay 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-L	N-by-L	Delay 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]

Output

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Port_1 — Delayed output
vector | matrix

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 Input	Delay Input	Output	Effect of Delay Input on Data Input
N (unoriented, one channel)	scalar	Unoriented (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-P	N-by-P	P 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-P	N-by-P	P taps. In addition, delay varies within each frame from sample to sample.
N-by-1 (one channel with frame size equal to N)	scalar	N-by-1	One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)	Unoriented (N)	N-by-1	Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)	N-by-1	N-by-1	Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)	1-by-P	N-by-P	P 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-P	N-by-P	P taps. In addition, delay varies within each frame from sample to sample.
N-by-L (L channels with frame size equal to N)	scalar	N-by-L	One delay value applied to all input channels
N-by-L (L channels with frame size equal to N)	1-by-L	N-by-L	Unique delay value for each input channel
N-by-L (L channels with frame size equal to N)	N-by-1	N-by-L	Delay 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-L	N-by-L	Delay 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-P	N-by-L-by-P	L 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-P	N-by-L-by-P	L channels. P taps per channel. Taps vary across channels.
N-by-L (L channels with frame size equal to N)	N-by-1-by-P	N-by-L-by-P	L 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-P	N-by-L-by-P	L 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 Input	Delay Input	Output	Effect of Delay Input on Data Input
N (unoriented, one channel)	scalar	Unoriented (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)	scalar	N-by-1	One delay value applied to the input channel
N-by-1 (one channel with frame size equal to N)	Unoriented (N)	N-by-1	Delay value varies within the frame from sample to sample
N-by-1 (one channel with frame size equal to N)	N-by-1	N-by-1	Delay value varies within the frame from sample to sample
N-by-L (L channels with N samples in each channel)	scalar	N-by-L	One delay value applied to all input channels
N-by-L (L channels with N samples in each channel)	1-by-L	N-by-L	Unique delay value for each input channel
N-by-L (L channels with N samples in each channel)	N-by-1	N-by-L	Delay 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-L	N-by-L	Delay 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]

Parameters

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Interpolation mode — Method of interpolation
`Linear` (default) | `FIR` | `Farrow`

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 D_max+1 most recent samples the In port receives for each channel. D_max is the value you specify in the Maximum delay (Dmax) in samples parameter.
FIR –– Polyphase FIR interpolation. In this mode, the block stores the D_max+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 D_max+ $\frac{N}{2}$ +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.

Interpolation filter half-length (P) — Half length of interpolation filter
`4` (default) | positive integer in the range [1 65535]

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.

Interpolation points per input sample — Number of interpolation points per input sample
`10` (default) | positive integer in the range [2, 65,535]

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.

Normalized input bandwidth (0 to 1) — Normalized input bandwidth
`1` (default) | real scalar in the range (0, 1]

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.

Farrow filter length (N) — Length of Farrow filter
`4` (default) | integer greater than or equal to 2

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.

Maximum delay (Dmax) in samples — Maximum delay
`100` (default) | integer in the range [0 65535]

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

Example: 200

Example: 500

Input processing — Method to process the input
`Columns as channels (frame based)` (default) | `Elements as channels (sample based)`

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 M_i 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 M_i-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 D_min ≤ v ≤ D_max.
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 D_min ≤ v ≤ D_max that specifies the number of sample intervals to delay each channel of the input, or it can be a scalar floating-point value, D_min ≤ v ≤ D_max, 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.

InitialConditions — Initial values in the memory
`0` (default) | scalar | 1-by-R-by-D array | 1-by-R-by-(D+L) array

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)

Disable direct feedthrough by increasing minimum possible delay by one — Disable direct feedthrough
`off` (default) | `on`

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.

For small input delay values — Action to take for small input delay values
`Clip to the minimum value necessary for centered kernel` (default) | `Use off-centered kernel` | `Switch to linear interpolation if kernel cannot be centered`

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 D_min 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 D_min 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 D_min 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.

Valid delay range (in samples) — Range of valid delay values
[0 100] (default) | [D_min D_max]

This parameter is read-only.

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

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

Linear –– D_min equal 0. D_max equals the value you specify in the Maximum delay (Dmax) in samples parameter.
FIR –– D_min equals P – 1, where P is the value you specify in Interpolation filter half-length (P). D_max equals the value you specify in the Maximum delay (Dmax) in samples parameter.
Farrow –– D_min equals N/2 – 1, where N is the value you specify in Farrow filter length (N). D_max 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

Rounding mode — Rounding method for fixed-point operations
`Zero` (default) | `Ceiling` | `Convergent` | `Floor` | `Nearest` | `Round` | `Simplest`

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.

Saturate on integer overflow — Method of overflow action
off (default) | on

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.

Coefficients — Data type of the coefficients
`Same word length as input` (default) | `Specify word length`

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 Data Types section.

Product output — Data type of the product output
`Same as first input` (default) | `Binary point scaling`

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 Data Types section.

Accumulator — Data type of accumulation operation
`Same as product output` (default) | `Same as first input` | `Binary point scaling`

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 Data Types section.

Product output polyval — Data type of the product polynomial value
`Same as first input` (default) | `Binary point scaling`

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 Data Types section.

Dependencies

This property applies when you set Interpolation mode to Farrow.

Accumulator polyval — Data type of the accumulator polynomial value
`Same as first input` (default) | `Binary point scaling`

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 Data Types section.

Dependencies

This property applies when you set Interpolation mode to Farrow.

Multiplicand polyval — Data type of multiplicand polynomial value
`Same as first input` (default) | `Binary point scaling`

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 Data Types section.

Dependencies

This property applies when you set Interpolation mode to Farrow.

Output — Data type of block output
`Same as accumulator` (default) | `Same as first input` | `Binary point scaling`

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 Data Types section.

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types
`off` (default) | `on`

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`

More About

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Fixed-Point Data Types

The diagrams in the following sections show the data types used within the Variable Fractional Delay for fixed-point signals.

Although you can specify most of these data types, the following data types are computed internally by the block and cannot be directly specified on the block dialog box.

Data Type Word Length Fraction Length

Data Type	Word Length	Fraction Length
v_f data type	Derived from the input to the Delay port. $v f W L = \max (D e l a y W L, D e l a y F L + c e i l (\log 2 (filterHalfLength)))$	Same as the fraction length of the input to the Delay port. $v f F L = D e l a y F L$
HoldInteger data type	Same word length as the input delay value	`0` bits
Integer data type	`32` bits	`0` bits

v_f data type

Derived from the input to the Delay port.

$v f W L = \max (D e l a y W L, D e l a y F L + c e i l (\log 2 (filterHalfLength)))$

Same as the fraction length of the input to the Delay port.

$v f F L = D e l a y F L$

HoldInteger data type Same word length as the input delay value 0 bits

Integer data type 32 bits 0 bits

Note

When the input is fixed point, all internal data types are signed fixed point.

To compute the integer (v_i) and fractional (v_f) parts of the input delay value (v), the Variable Fractional Delay block uses the following equations:

$\begin{array}{l} D_{m i n} < v < D_{m a x} \Rightarrow {\begin{cases} v_{i} = floor (v) \\ v_{f} = v - v i \end{cases} \\ v \leq D_{m i n} \Rightarrow {\begin{cases} v_{i} = D_{m i n} \\ v_{f} = 0 \end{cases} \\ v \geq D_{m a x} \Rightarrow {\begin{cases} v_{i} = D_{m a x} \\ v_{f} = 0 \end{cases} \end{array}$

Linear Interpolation Mode

The following diagram shows the fixed-point data types used by the Linear interpolation mode of the Variable Fractional Delay block.

FIR Interpolation Mode

The following diagram illustrates how the Variable Fractional Delay block selects the arm of the polyphase filter structure that most closely matches the fractional delay value (v_f).

The following diagram shows the fixed-point data types used by the variable fractional delay algorithm in the FIR interpolation mode.

You can set the coefficient, product output, accumulator, and output data types in the block. This diagram shows that input data is stored in the input buffer with the same data type and scaling as the input. The block stores filtered data and any initial conditions in the output buffer using the output data type and scaling that you set.

When at least one of the inputs to the multiplier is real, the output of the multiplier is in the product output data type. When both inputs to the multiplier are complex, the result of the multiplication is in the accumulator data type. For details on the complex multiplication, see Multiplication Data Types.

Farrow Interpolation Mode

The following diagram shows the fixed-point data types used by the Farrow interpolation mode when:

Farrow filter length is set to 4
Farrow small delay action is set to Clip to the minimum value necessary for centered kernel

The following diagram shows the fixed-point data types used by the Farrow interpolation mode when:

Farrow filter length is set to 4.
Farrow small delay action is set to Use off-centered kernel.

Diff is computed from the integer part of the delay value (v_i) and the farrow filter length (N) according to the following equation:

$\begin{array}{l} D i f f = v_{i} - (\frac{N - 1}{2}) \\ D i f f \geq 0 \Rightarrow D i f f = 0 \\ D i f f < 0 \Rightarrow D i f f = - D i f f \end{array}$

The following diagram shows the fixed-point data types used by the Digital Filter's FIR direct form filter.

Algorithms

expand all

The delay value specified at the Delay port serves as an index into the block's memory, U, which stores, at a minimum, the D_max+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.

Linear Interpolation Mode

For noninteger delays, at each sample time, the linear interpolation method uses the two samples in memory nearest to the specified delay to compute a value for the sample at that time.

For a vector data input, the output vector, y, is computed using the following relation:

vi = floor(v)	
vf = v-vi					
y(i) = U(i-vi-1)*vf + U(i-vi)*(1-vf)

where,

i –– Index of the current sample
v –– Fractional delay
vi –– Integer part of the delay
vf –– Fractional part of the delay
U –– Input data vector
y –– Output data vector
U(i-vi), U(i-vi-1) –– Two samples in memory nearest to the specified delay
i-vi –– Distance, in samples, between the current index and the nearest point in the interpolation line.

The variable fractional delay stores the Dmax+1 most recent samples received at the input for each channel, where Dmax is the maximum delay specified. U represents the stored samples.

FIR Interpolation Mode

In the FIR interpolation mode, the block stores the Dmax+P+1 most recent samples received at the input for each channel, where P is the specified interpolation filter half-length.

In this mode, the block provides a discrete set of fractional delays:

$v + \frac{i}{L}, v \geq P - 1, i = 0, 1, ..., L - 1$

If v is less than P – 1, the behavior depends on the For small input delay values parameter. You can specify the block's behavior when the input delay value is too small to center the kernel (less than P-1), by setting the For small input delay values parameter:

Clip to the minimum value necessary for centered kernel –– The FIR interpolation method remains in use. The small input delay values are clipped to the smallest value necessary to center the kernel.
Switch to linear interpolation if kernel cannot be centered –– Fractional delays are computed using linear interpolation when the input delay value is less than P-1.

In the FIR interpolation mode, the algorithm implements a polyphase structure to compute a value for each sample at the specified delay. Each arm of the structure corresponds to a different delay value. The output computed for each sample corresponds to the output of the arm with a delay value nearest to the specified input delay. Therefore, only a discrete set of delays is actually possible. The number of coefficients in each of the L filter arms of the polyphase structure is 2P. In most cases, using values of P between 4 and 6 provides you with reasonably accurate interpolation values.

The designMultirateFIR function designs the FIR interpolation filter.

For example, when you set the following values:

Interpolation filter half-length (P) to 4
Interpolation points per input sample to 10
Normalized input bandwidth to 1
Stopband attenuation to 80 dB

The filter coefficients are given by:

b = designMultirateFIR(10,1,4,80);

The algorithm then implements this filter as a polyphase structure.

Increasing the filter half length (P) increases the accuracy of the interpolation, but also increases the number of computations performed per input sample. The amount of memory needed to store the filter coefficients increases too. Increasing the interpolation points per sample (L) increases the number of representable discrete delay points, but also increases the simulation's memory requirements. The computational load per sample is not affected.

The normalized input bandwidth from 0 to 1 allows you to take advantage of the bandlimited frequency content of the input. For example, if you know that the input signal does not have frequency content above F_s/4, you can specify 0.5 normalized bandwidth to constrain the frequency content of the output to that range.

Note

You can consider each of the L interpolation filters to correspond to one output phase of an upsample-by-L FIR filter. Therefore, the normalized input value improves the stopband in critical regions and relaxes the stopband requirements in frequency regions without signal energy.

Farrow Interpolation Mode

In the farrow interpolation mode, the block stores the Dmax+N/2+1 most recent samples received at the input for each channel, where N is the specified farrow filter length.

The algorithm uses the LaGrange method to interpolate values.

To increase the minimum possible delay value, select the Disable direct feedthrough by increasing minimum possible delay by one check box. Checking this box prevents algebraic loops from occurring when you use the block inside a feedback loop.

To specify the behavior when the input delay value is too small to center the kernel (less than $\frac{N}{2}$ – 1), use the Farrow small delay action setting.

Clip to the minimum value necessary for centered kernel –– The block clips small input delay values to the smallest value necessary to keep the kernel centered. This increases D_min but yields more accurate interpolation values.
Use off-centered kernel –– The fractional delays are computed using a Farrow filter with an off-centered kernel. This mode does not increase D_min, but the results for input delay values less than $\frac{N}{2}$ – 1 are less accurate than the results achieved by keeping the kernel centered.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

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

Version History

Introduced before R2006a

Variable Fractional Delay

Description

Examples

Delay Signal Using Multitap Fractional Delay

Ports

Input

In — Data input vector | matrix

Delay — Delay input scalar | vector | matrix | N-D array

Output

Port_1 — Delayed output vector | matrix

Parameters

Interpolation mode — Method of interpolation Linear (default) | FIR | Farrow

Interpolation filter half-length (P) — Half length of interpolation filter 4 (default) | positive integer in the range [1 65535]

Dependencies

Interpolation points per input sample — Number of interpolation points per input sample 10 (default) | positive integer in the range [2, 65,535]

Dependencies

Normalized input bandwidth (0 to 1) — Normalized input bandwidth 1 (default) | real scalar in the range (0, 1]

Dependencies

Farrow filter length (N) — Length of Farrow filter 4 (default) | integer greater than or equal to 2

Dependencies

Maximum delay (Dmax) in samples — Maximum delay 100 (default) | integer in the range [0 65535]

Input processing — Method to process the input Columns as channels (frame based) (default) | Elements as channels (sample based)

InitialConditions — Initial values in the memory 0 (default) | scalar | 1-by-R-by-D array | 1-by-R-by-(D+L) array

Disable direct feedthrough by increasing minimum possible delay by one — Disable direct feedthrough off (default) | on

For small input delay values — Action to take for small input delay values Clip to the minimum value necessary for centered kernel (default) | Use off-centered kernel | Switch to linear interpolation if kernel cannot be centered

Dependencies

Valid delay range (in samples) — Range of valid delay values [0 100] (default) | [Dmin Dmax]

Fixed-Point Properties

Rounding mode — Rounding method for fixed-point operations Zero (default) | Ceiling | Convergent | Floor | Nearest | Round | Simplest

Saturate on integer overflow — Method of overflow action off (default) | on

Coefficients — Data type of the coefficients Same word length as input (default) | Specify word length

Product output — Data type of the product output Same as first input (default) | Binary point scaling

Accumulator — Data type of accumulation operation Same as product output (default) | Same as first input | Binary point scaling

Product output polyval — Data type of the product polynomial value Same as first input (default) | Binary point scaling

Dependencies

Accumulator polyval — Data type of the accumulator polynomial value Same as first input (default) | Binary point scaling

Dependencies

Multiplicand polyval — Data type of multiplicand polynomial value Same as first input (default) | Binary point scaling

Dependencies

Output — Data type of block output Same as accumulator (default) | Same as first input | Binary point scaling

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types off (default) | on

Block Characteristics

More About

Fixed-Point Data Types

Algorithms

Linear Interpolation Mode

FIR Interpolation Mode

Farrow Interpolation Mode

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

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

Version History

See Also

Objects

Blocks

Topics

In — Data input
vector | matrix

Delay — Delay input
scalar | vector | matrix | N-D array

Port_1 — Delayed output
vector | matrix

Interpolation mode — Method of interpolation
`Linear` (default) | `FIR` | `Farrow`

Interpolation filter half-length (P) — Half length of interpolation filter
`4` (default) | positive integer in the range [1 65535]

Interpolation points per input sample — Number of interpolation points per input sample
`10` (default) | positive integer in the range [2, 65,535]

Normalized input bandwidth (0 to 1) — Normalized input bandwidth
`1` (default) | real scalar in the range (0, 1]

Farrow filter length (N) — Length of Farrow filter
`4` (default) | integer greater than or equal to 2

Maximum delay (Dmax) in samples — Maximum delay
`100` (default) | integer in the range [0 65535]

Input processing — Method to process the input
`Columns as channels (frame based)` (default) | `Elements as channels (sample based)`

InitialConditions — Initial values in the memory
`0` (default) | scalar | 1-by-R-by-D array | 1-by-R-by-(D+L) array

Disable direct feedthrough by increasing minimum possible delay by one — Disable direct feedthrough
`off` (default) | `on`

For small input delay values — Action to take for small input delay values
`Clip to the minimum value necessary for centered kernel` (default) | `Use off-centered kernel` | `Switch to linear interpolation if kernel cannot be centered`

Valid delay range (in samples) — Range of valid delay values
[0 100] (default) | [D_min D_max]

Rounding mode — Rounding method for fixed-point operations
`Zero` (default) | `Ceiling` | `Convergent` | `Floor` | `Nearest` | `Round` | `Simplest`

Saturate on integer overflow — Method of overflow action
off (default) | on

Coefficients — Data type of the coefficients
`Same word length as input` (default) | `Specify word length`

Product output — Data type of the product output
`Same as first input` (default) | `Binary point scaling`

Accumulator — Data type of accumulation operation
`Same as product output` (default) | `Same as first input` | `Binary point scaling`

Product output polyval — Data type of the product polynomial value
`Same as first input` (default) | `Binary point scaling`

Accumulator polyval — Data type of the accumulator polynomial value
`Same as first input` (default) | `Binary point scaling`

Multiplicand polyval — Data type of multiplicand polynomial value
`Same as first input` (default) | `Binary point scaling`

Output — Data type of block output
`Same as accumulator` (default) | `Same as first input` | `Binary point scaling`

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types
`off` (default) | `on`

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

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