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
antiimaging filter, which is usually an approximation of an ideal bandlimited
interpolation filter. To design an FIR antiimaging 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 discretetime signal has a sample rate that is L times the original sample rate.
The actual block algorithm implements a directform 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 singlerate 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.
This block supports variablesize inputs. That is, the frame size (number of rows) of the signal can change during simulation but the number of channels cannot.
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 =
[b_{0},
b_{1}, …,
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 antiimaging 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 singlerate 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.
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
— 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].
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 = [b_{0},
b_{1}, …,
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 antiimaging 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
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 singlerate
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 singlerate 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 toColumns 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 

 L samples 
 L frames (K_{i} 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.
Rounding mode
— Rounding mode
Floor
(default)  Ceiling
 Convergent
 Nearest
 Round
 Simplest
 Zero
Select the rounding
mode for fixedpoint 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 fullprecision mode.
Saturate on integer overflow
— Saturate on integer overflow
off
(default)  on
When you select this parameter, the block saturates the result of its
fixedpoint operation. When you clear this parameter, the block wraps the
result of its fixedpoint operation. For details on
saturate
and wrap
, see overflow
mode for fixedpoint 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 fullprecision 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)
orfixdt(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
fixedpoint 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
fixedpoint 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 fixedpoint 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 fixedpoint 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 fixedpoint tools
— Prevent fixedpoint tools from overriding data types
off
(default)  on
Select this parameter to prevent the fixedpoint tools from overriding the data types you specify in the block dialog box.
Model Examples
Block Characteristics
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

More About
FrameBased 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 singlerate 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 singlerate 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 singlerate FIR Interpolation, see FIR Interpolation Using SingleRate 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}byN 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 FrameBased Processing for an example that uses the FIR Interpolation block in this mode.
SampleBased Processing
When you set the Input processing parameter to
Elements as channels (sample based)
, the block treats
a PbyQ 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 singlerate processing
, the FIR Interpolation
block always has zerotasking latency. Zerotasking 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 

 L samples 
 L frames (K_{i} 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 samplebased 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 framebased 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 TimeBased 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:
$$H(z)=\begin{array}{c}\left({b}_{0}+{b}_{L}{z}^{L}+{b}_{2L}{z}^{2L}+\mathrm{..}+{b}_{NL+1}{z}^{(NL+1)}\right)+\\ {z}^{1}\left({b}_{1}+{b}_{L+1}{z}^{L}+{b}_{2L+1}{z}^{2L}+\mathrm{..}+{b}_{NL+2}{z}^{(NL+1)}\right)+\\ \begin{array}{c}\vdots \\ {z}^{(L1)}\left({b}_{L1}+{b}_{2L1}{z}^{L}+{b}_{3L1}{z}^{2L}+\mathrm{..}+{b}_{N}{z}^{(NL+1)}\right)\end{array}\end{array}$$
L is the number of polyphase components, and its value equals the interpolation factor that you specify.
You can write this equation as:
$$H(z)={E}_{0}({z}^{L})+{z}^{1}{E}_{1}({z}^{L})+\mathrm{...}+{z}^{(L1)}{E}_{L1}({z}^{L})$$
E_{0}(z^{L}), E_{1}(z^{L}), ..., E_{L1}(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: PrenticeHall, 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 singlerate processing
.Input signal is realvalued with real filter coefficients.
Input signal is complexvalued with real or complex filter coefficients.
Input signal has a data type of
single
ordouble
.
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.
To reduce area or increase speed, the FIR Decimator block supports blocklevel optimizations.
Rightclick on the block or the subsystem to open the corresponding HDL Properties dialog box and set optimization properties.
Serial Architectures  When you select 
Distributed Arithmetic  Distributed Arithmetic properties
DALUTPartition and
DARadix are supported for the

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 treebased adder, for a latency of

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 shiftandadd logic. When you choose a fully
parallel filter implementation, you can set CoeffMultipliers to

DALUTPartition  Specify distributed arithmetic partialproduct 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 ( 
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 cascadeserial 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). 
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

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

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

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 fixedpoint 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.
FixedPoint Conversion
Design and simulate fixedpoint systems using FixedPoint Designer™.
The following diagram shows the data types used within the FIR Interpolation block for fixedpoint 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
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