FIR Interpolation

Upsample and filter input signals

• Library:

• DSP System Toolbox / Filtering / Multirate Filters

  DSP System Toolbox HDL Support / Filtering

• 

Description

The FIR Interpolation block upsamples an input by the integer upsampling factor L along the first dimension. The FIR interpolator (as shown in the schematic) conceptually consists of an upsampler followed by an FIR anti-imaging filter, which is usually an approximation of an ideal band-limited interpolation filter. To design an FIR anti-imaging filter, use the designMultirateFIR function.

The upsampler upsamples each channel of the input to a higher rate by inserting L–1 zeros between samples. The FIR filter that follows filters each channel of the upsampled data. The resulting discrete-time signal has a sample rate that is L times the original sample rate.

The actual block algorithm implements a direct-form FIR polyphase structure, an efficient equivalent of the combined system depicted in the diagram. For more details, see Algorithms.

You can use the FIR Interpolation block inside triggered subsystems when you set the Rate options parameter to Enforce single-rate processing.

Under specific conditions, this block also supports SIMD code generation. For more details, see Code Generation.

Ports

Input

In — Data input

vector | matrix

Specify the data input as a vector or a matrix.

When you set Input processing to Columns as channels (frame based) and Rate options to Enforce single-rate processing, the input can be a variable-size signal. That is, the frame size (number of rows) and the number of channels (columns) of the signal can change during simulation.

This port is unnamed until you set Coefficient source to Input port.

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

Num — Numerator coefficients

vector

Specify the numerator coefficients of the FIR filter as a vector.

The transfer function H(z) of the FIR filter is given by:

$$H(z)=b_0+b_1{z}^{-1}+\mathrm{...}+b_N{z}^{-N}$$

You can generate the FIR filter coefficient vector, b = [b₀, b₁, …, b_N], using one of the DSP System Toolbox™ filter design functions such as designMultirateFIR, firnyquist, firhalfband, firgr or firceqrip.

To act as an effective anti-imaging filter, the coefficients usually correspond to a lowpass filter with a normalized cutoff frequency no greater than the reciprocal of the interpolation factor. To design such a filter, use the designMultirateFIR function.

Coefficient values are tunable. That is, their values can change during simulation while their properties such as size, data type, and complexity cannot change.

The data type of the Num input must match the data type of the In input.

Dependencies

The Num input port appears when you set Coefficient source as Input port.

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

Output

Out — Interpolator output

vector | matrix

Output of the FIR Interpolator block, returned as a vector or a matrix.

When Rate options is set to:

• Enforce single-rate processing — When you select this option, the block maintains the input sample rate, and interpolates the signal by increasing the output frame size by a factor of L.

• Allow multirate processing — When you select this option, the block interpolates the signal such that the output sample rate is L times faster than the input sample rate.

When the input is a variable-size signal, the output is also a variable-size signal.

This port is unnamed until you set Coefficient source to Input port.

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

Parameters

Coefficient source

Coefficient source — FIR filter coefficient source

Auto (default) | Dialog parameters | Input port | Filter object

Specify the FIR filter coefficient source as one of the following:

• Dialog parameters –– Specify the filter coefficients through the FIR filter coefficients parameter in the block dialog box.

• Input port –– Specify the filter coefficients through the Num input port.

• Filter object –– Specify the filter using a dsp.FIRInterpolator System object™.

• Auto –– When you select Auto, the block designs an FIR interpolator using the interpolation factor you specify in Interpolation factor. The designMultirateFIR function designs the filter and returns the coefficients used by the block.

  For more information on the filter design, see Orfanidis [2].

Main Tab

FIR filter coefficients — Lowpass FIR filter coefficients

designMultirateFIR(3,1) (default) | vector

Specify the numerator coefficients of the FIR filter transfer function H(z).

$$H(z)=b_0+b_1{z}^{-1}+\mathrm{...}+b_N{z}^{-N}$$

You can generate the FIR filter coefficient vector, b = [b₀, b₁, …, b_N], using one of the DSP System Toolbox filter design functions such as designMultirateFIR, firnyquist, firhalfband, firgr or firceqrip.

To act as an effective anti-imaging filter, the coefficients usually correspond to a lowpass filter with a normalized cutoff frequency no greater than the reciprocal of the interpolation factor. To design such a filter, use the designMultirateFIR function.

The block internally initializes all filter states to zero.

Dependencies

This parameter appears only when you set the Coefficient source to Dialog parameters.

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

Interpolation factor — Interpolation factor

3 (default) | positive scalar

Specify the integer factor L. The block increases the sample rate of the input sequence by this factor.

Dependencies

This parameter appears only when you set the Coefficient source to Dialog parameters, Input port, or Auto.

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

Filter object — Filter object

dsp.FIRInterpolator

Specify the name of the multirate filter object that you want the block to implement. You must specify the filter as a dsp.FIRInterpolator System object.

You can define the System object directly in the block dialog box. Alternatively, you can define the object in a MATLAB^® workspace variable and specify the variable in the block dialog box.

For information on creating System objects, see Define Basic System Objects.

Dependencies

This parameter appears only when you set the Coefficient source to Filter object.

Input processing — Method to process input signals

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) — When you select this option, the block treats each column of the input as a separate channel.

• Elements as channels (sample based) — When you select this option, the block treats each element of the input as a separate channel.

Rate options — Method by which block interpolates input

Enforce single-rate processing (default) | Allow multirate processing

Specify the method by which the block should interpolate the input. You can select one of the following options:

• Enforce single-rate processing — When you select this option, the block maintains the input sample rate, and interpolates the signal by increasing the output frame size by a factor of L. To select this option, you must set the Input processing parameter to Columns as channels (frame based).

• Allow multirate processing — When you select this option, the block interpolates the signal such that the output sample rate is L times faster than the input sample rate.

Output buffer initial conditions — Initial conditions

0 (default) | scalar | matrix

When you set the Rate options parameter to Allow multirate processing and run your models in Simulink^® MultiTasking mode, the block exhibits latency. The amount of latency for multirate, multitasking operation depends on how you set the Input processing parameter.

Input processing	Latency
Elements as channels (sample based)	L samples
Columns as channels (frame based)	L frames (Ki samples per frame)

When the block exhibits latency, the default initial condition is zero. Alternatively, you can use the Output buffer initial conditions parameter to specify a matrix of initial conditions containing one value for each channel or a scalar initial condition that the block applies to all channels. The block divides the Output buffer initial conditions by the Interpolation factor and outputs the scaled initial conditions until the first filtered input sample becomes available.

Output buffer initial conditions are stored in the output data type and scaling.

See Latency for more information about latency in the FIR Interpolation block.

Dependencies

This parameter appears only when you configure the block to perform multirate processing by setting Rate options to Allow multirate processing.

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

View Filter Response — View filter response

button

Click on this button to open the Filter Visualization Tool (fvtool) and display the filter response of the filter defined in the block dialog box.

Data Types Tab

Rounding mode — Rounding mode

Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero

Select the rounding mode for fixed-point operations. The default is Floor. The filter coefficients do not obey this parameter and always round to Nearest.

Note

The Rounding mode and Saturate on integer overflow settings have no effect on numerical results when all the following conditions exist:

• Product output is Inherit: Inherit via internal rule

• Accumulator is Inherit: Inherit via internal rule

• Output is Inherit: Same as accumulator

With these data type settings, the block is effectively operating in the full-precision mode.

Saturate on integer overflow — Saturate on integer overflow

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.

Note

The Rounding mode and Saturate on integer overflow parameters have no effect on numeric results when all these conditions are met:

• Product output data type is Inherit: Inherit via internal rule.

• Accumulator data type is Inherit: Inherit via internal rule.

With these data type settings, the block operates in the full-precision mode.

Coefficients Data Type — Coefficients data type

Inherit: Same word length as input (default) | fixdt(1,16) | fixdt(1,16,0)

Specify the coefficients data type. See Fixed Point and Multiplication Data Types for illustrations depicting the use of the coefficients data type in this block.

You can set this parameter to one of the following:

• Inherit: Same word length as input

• fixdt(1,16,0) or fixdt(1,16) –– Specify a data type object.

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Coefficients parameter.

See Specify Data Types Using Data Type Assistant (Simulink) for more information.

Dependencies

This parameter appears only when you set Coefficient source to Dialog parameters, Filter object, or Auto.

When Coefficient source is set to Filter object, Coefficients parameter is automatically set to Same word length as input.

Coefficients Minimum — Minimum value of filter coefficients

[] (default) | scalar

Specify the minimum value of the filter coefficients. The default value is [] (unspecified). Simulink software uses this value to perform automatic scaling of fixed-point data types.

Dependencies

This parameter appears only when you set Coefficient source to Dialog parameters or Auto.

Coefficients Maximum — Maximum value of filter coefficients

[] (default) | scalar

Specify the maximum value of the filter coefficients. The default value is [] (unspecified). Simulink software uses this value to perform automatic scaling of fixed-point data types.

Dependencies

This parameter appears only when you set Coefficient source to Dialog parameters or Auto.

Product output Data Type — Product output data type

Inherit: Inherit via internal rule (default) | Inherit: Same as input | fixdt(1,16,0)

Specify the product output data type. See Fixed Point and Multiplication Data Types for illustrations depicting the use of the product output data type in this block.

You can set this parameter to one of the following:

• Inherit: Inherit via internal rule

  For more information on this rule, see Inherit via Internal Rule.

• Inherit: Same as input

• fixdt(1,16,0) –– Specify a data type object.

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Product output parameter.

See Specify Data Types Using Data Type Assistant (Simulink) for more information.

Dependencies

When Coefficient source is set to Filter object, Product output parameter is automatically set to Full precision.

Accumulator Data Type — Accumulator data type

Inherit: Inherit via internal rule (default) | Inherit: Same as input | Inherit: Same as product output | fixdt(1,16,0)

Specify the accumulator data type. See Fixed Point for illustrations depicting the use of the accumulator data type in this block.

You can set this parameter to one of the following:

• Inherit: Inherit via internal rule.

  For more information on this rule, see Inherit via Internal Rule.

• Inherit: Same as input

• Inherit: Same as product output

• fixdt(1,16,0) –– Specify a data type object.

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Accumulator parameter.

See Specify Data Types Using Data Type Assistant (Simulink) for more information.

Dependencies

When Coefficient source is set to Filter object, Accumulator parameter is automatically set to Full precision.

Output Data Type — Output data type

Inherit: Same as accumulator (default) | Inherit: Same as input | Inherit: Same as product output | fixdt(1,16,0)

Specify the output data type. See Fixed Point for illustrations depicting the use of the output data type in this block.

You can set it to one of the following:

• Inherit: Same as accumulator

• Inherit: Same as input

• Inherit: Same as product output

• fixdt(1,16,0) –– Specify a data type object.

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Output parameter.

See Control Data Types of Signals (Simulink) for more information.

Dependencies

When Coefficient source is set to Filter object, Output parameter is automatically set to Same as accumulator.

Output Minimum — Minimum value of block output

[] (default) | scalar

Specify the minimum value that the block should output. The default value is [] (unspecified). Simulink software uses this value to perform:

• Simulation range checking (see Specify Signal Ranges (Simulink))

• Automatic scaling of fixed-point data types

Dependencies

This parameter appears only when you set Coefficient source to Dialog parameters, Input port, or Auto.

Output Maximum — Maximum value of block output

[] (default) | scalar

Specify the maximum value that the block should output. The default value is [] (unspecified). Simulink software uses this value to perform:

• Simulation range checking (see Specify Signal Ranges (Simulink))

• Automatic scaling of fixed-point data types

Dependencies

This parameter appears only when you set Coefficient source to Dialog parameters, Input port, or Auto.

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.

Model Examples

FIR Interpolation Using Single-Rate Processing

Use FIR Interpolation block in single-rate processing mode.

Open Model

FIR Interpolation Using Multirate Frame-Based Processing

Use FIR Interpolation block in multirate frame-based processing mode.

Open Model

Polyphase Implementation of FIR Interpolation Block

Polyphase implementation of the FIR interpolator.

Open Model

Two-Stage Multirate Narrow Lowpass Filter

Implement a two-stage multirate narrow band lowpass filter using FIR Decimation and FIR Interpolation blocks.

Open Model

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

Frame-Based Processing

When you set the Input processing parameter to Columns as channels (frame based), the block resamples each column of the input over time. In this mode, the block can perform either single-rate or multirate processing. You can use the Rate options parameter to specify how the block resamples the input:

• When you set the Rate options parameter to Enforce single-rate processing, the input and output of the block have the same sample rate. To interpolate the output while maintaining the input sample rate, the block resamples the data in each column of the input such that the frame size of the output (K_o) is L times larger than that of the input (K_o = K_i*L).

  For an example of single-rate FIR Interpolation, see FIR Interpolation Using Single-Rate Processing.

• When you set the Rate options parameter to Allow multirate processing, the input and output of the FIR Interpolation block are the same size. However, the sample rate of the output is L times faster than that of the input. In this mode, the block treats a K_i-by-N matrix input as N independent channels. The block interpolates each column of the input over time by keeping the frame size constant (K_i=K_o), while making the output frame period (T_fo) L times shorter than the input frame period (T_fo = T_fi/L).

  See FIR Interpolation Using Multirate Frame-Based Processing for an example that uses the FIR Interpolation block in this mode.

Sample-Based Processing

When you set the Input processing parameter to Elements as channels (sample based), the block treats a P-by-Q matrix input as P*Q independent channels, and interpolates each channel over time. The output sample period (T_so) is L times shorter than the input sample period (T_so = T_si/L), while the input and output sizes remain identical.

Latency

When you run your models in the Simulink SingleTasking mode or set the Input processing parameter to Columns as channels (frame based) and the Rate options parameter to Enforce single-rate processing, the FIR Interpolation block always has zero-tasking latency. Zero-tasking latency means that the block propagates the first filtered input sample (received at time t=0) as the first output sample. That first output sample is then followed by L–1 interpolated values, the second filtered input sample, and so on.

The only time the FIR Interpolation block exhibits latency is when you set the Rate options parameter set to Allow multirate processing and run your models in the Simulink MultiTasking mode. The amount of latency for a multirate, multitasking operation depends on how you set the Input processing parameter.

Input processing	Latency
Elements as channels (sample based)	L samples
Columns as channels (frame based)	L frames (Ki samples per frame)

When the block exhibits latency, the default initial condition is zero. Alternatively, you can use the Output buffer initial conditions parameter to specify a matrix of initial conditions containing one value for each channel or a scalar initial condition that the block applies to all channels. The block scales the Output buffer initial conditions by the Interpolation factor and outputs the scaled initial conditions until the first filtered input sample becomes available.

When the block is in the sample-based processing mode, the block outputs the scaled initial conditions at the start of each channel, followed immediately by the first filtered input sample, then L–1 interpolated values, and so on.

When the block is in the frame-based processing mode and using the default initial condition of zero, the first K_i*L output rows contain zeros, where K_i is the input frame size. The first filtered input sample (first filtered row of the input matrix) appears in the output as sample K_i*L+1. That value is then followed by L–1 interpolated values, the second filtered input sample, and so on.

Note

For more information on latency and the Simulink tasking modes, see Excess Algorithmic Delay (Tasking Latency) and Time-Based Scheduling and Code Generation (Simulink Coder).

Algorithms

The FIR interpolation filter is implemented efficiently using a polyphase structure.

To derive the polyphase structure, start with the transfer function of the FIR filter:

$$H(z)=b_0+b_1{z}^{-1}+\mathrm{...}+b_N{z}^{-N}$$

N+1 is the length of the FIR filter.

You can rearrange this equation as follows:

$$H(z)=\begin{array}{c}\left(b_0+b_L{z}^{-L}+b_{2L}{z}^{-2L}+\mathrm{..}+b_{N-L+1}{z}^{-(N-L+1)}\right)+\\ z^{-1}\left(b_1+b_{L+1}{z}^{-L}+b_{2L+1}{z}^{-2L}+\mathrm{..}+b_{N-L+2}{z}^{-(N-L+1)}\right)+\\ \begin{array}{c}⋮\\ z^{-(L-1)}\left(b_{L-1}+b_{2L-1}{z}^{-L}+b_{3L-1}{z}^{-2L}+\mathrm{..}+b_N{z}^{-(N-L+1)}\right)\end{array}\end{array}$$

L is the number of polyphase components, and its value equals the interpolation factor that you specify.

You can write this equation as:

$$H(z)=E_0(z^L)+z^{-1}{E}_1(z^L)+\mathrm{...}+z^{-(L-1)}{E}_{L-1}(z^L)$$

E₀(z^L), E₁(z^L), ..., E_L-1(z^L) are polyphase components of the FIR filter H(z).

Conceptually, the FIR interpolation filter contains an upsampler followed by an FIR lowpass filter H(z).

Replace H(z) with its polyphase representation.

Here is the multirate noble identity for interpolation.

Applying the noble identity for interpolation moves the upsampling operation to after the filtering operation. This move enables you to filter the signal at a lower rate.

You can replace the upsampling operator, delay block, and adder with a commutator switch. The switch starts on the first branch 0 and moves in the counterclockwise direction, each time receiving one sample from each branch. The interpolator effectively outputs L samples for every one input sample it receives. Hence the sample rate at the output of the FIR interpolation filter is Lfs.

References

[1] Fliege, N. J. Multirate Digital Signal Processing: Multirate Systems, Filter Banks, Wavelets . West Sussex, England: John Wiley & Sons, 1994.

[2] 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 Simulink® Coder™.

Generated code relies on the memcpy or memset function (string.h) under certain conditions.

The FIR Interpolation block supports SIMD code generation using Intel AVX2 technology under these conditions:

• Input processing is set to Columns as channels (frame based).

• Rate options is set to Enforce single-rate processing.

• Input signal is real-valued with real filter coefficients.

• Input signal is complex-valued with real or complex filter coefficients.

• Input signal has a data type of single or double.

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

HDL Code Generation

Generate Verilog and VHDL code for FPGA and ASIC designs using HDL Coder™.

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

HDL Coder supports Coefficient source options Dialog parameters, Filter object, or Auto.

Block Optimizations

To reduce area or increase speed, the FIR Decimator block supports block-level optimizations.

Right-click on the block or the subsystem to open the corresponding HDL Properties dialog box and set optimization properties.

Serial Architectures	When you select Fully Serial architecture, the SerialPartition property is set on the FIR Interpolation Block.
Distributed Arithmetic	Distributed Arithmetic properties DALUTPartition and DARadix are supported for the Distributed Arithmetic (DA) architecture with a default FIR filter structure. structures.
Pipelining	When you use AddPipelineRegisters, registers are placed based on the filter structure. The pipeline register placement determines the latency. A pipeline register is added between levels of a tree-based adder, for a latency of ceil(log2(PL))-1, where PL is polyphase filter length.

HDL Filter Properties

AddPipelineRegisters	Insert a pipeline register between stages of computation in a filter. See also AddPipelineRegisters (HDL Coder).
CoeffMultipliers	Specify the use of canonical signed digit (CSD) optimization to decrease filter area by replacing coefficient multipliers with shift-and-add logic. When you choose a fully parallel filter implementation, you can set CoeffMultipliers to csd or factored-csd. The default is multipliers, which retains multipliers in the HDL. See also CoeffMultipliers (HDL Coder).
DALUTPartition	Specify distributed arithmetic partial-product LUT partitions as a vector of the sizes of each partition. The sum of all vector elements must be equal to the filter length. The maximum size for a partition is 12 taps. Set DALUTPartition to a scalar value equal to the filter length to generate DA code without LUT partitions. See also DALUTPartition (HDL Coder).
DARadix	Specify how many distributed arithmetic bit sums are computed in parallel. A DA radix of 8 (2^3) generates a DA implementation that computes three sums at a time. The default value is 2^1, which generates a fully serial DA implementation. See also DARadix (HDL Coder).
MultiplierInputPipeline	Specify the number of pipeline stages to add at filter multiplier inputs. See also MultiplierInputPipeline (HDL Coder).
MultiplierOutputPipeline	Specify the number of pipeline stages to add at filter multiplier outputs. See also MultiplierOutputPipeline (HDL Coder).
SerialPartition	Specify partitions for partly serial or cascade-serial filter implementations as a vector of the lengths of each partition. For a fully serial implementation, set this parameter to the length of the filter. See also SerialPartition (HDL Coder).

HDL Block Properties

ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For more details, see ConstrainedOutputPipeline (HDL Coder).
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see OutputPipeline (HDL Coder).

Restrictions

• You must set Initial conditions to zero. HDL code generation is not supported for nonzero initial states.

• Vector and frame inputs are not supported for HDL code generation.

• When you select Dialog parameters, the following fixed-point options are not supported for HDL code generation:

  • Coefficients: Slope and Bias scaling

• CoeffMultipliers options are supported only when using a fully parallel architecture. When you select a serial architecture, CoeffMultipliers is hidden from the HDL Block Properties dialog box.

Fixed-Point Conversion

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

The following diagram shows the data types used within the FIR Interpolation block for fixed-point signals.

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 in the block dialog box.

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 performed by this block, see Multiplication Data Types.

Note

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

Version History

Introduced before R2006a