dsp.IIRHalfbandInterpolator
Interpolate by a factor of two using polyphase IIR
Description
The dsp.IIRHalfbandInterpolator
System object™ performs efficient polyphase interpolation of the input signal by a factor of
two. To design the halfband filter, you can specify the object to use an elliptic design or a
quasi-linear phase design. The object uses these design methods to compute the filter
coefficients. To filter the inputs, the object uses a polyphase structure. The allpass filters
in the polyphase structure are in a minimum multiplier form.
Elliptic design introduces nonlinear phase and creates the filter using fewer coefficients than quasi linear design. Quasi-linear phase design overcomes phase nonlinearity at the cost of additional coefficients.
Alternatively, instead of designing the halfband filter using a design method, you can specify the filter coefficients directly. When you choose this option, the allpass filters in the two branches of the polyphase implementation can be in a minimum multiplier form or in a wave digital form.
You can also use dsp.IIRHalfbandInterpolator
object to implement the
synthesis portion of a two-band filter bank to synthesize a signal from lowpass and highpass
subbands.
To upsample and interpolate your data:
Create the
dsp.IIRHalfbandInterpolator
object and set its properties.Call the object with arguments, as if it were a function.
To learn more about how System objects work, see What Are System Objects?
Creation
Syntax
Description
iirhalfbandinterp = dsp.IIRHalfbandInterpolator
returns an IIR
halfband interpolation filter, iirhalfbandinterp
, with the default
settings. Under the default settings, the System object upsamples and interpolates the input data using a halfband frequency of
22050
Hz, a transition width of 4100
Hz, and a
stopband attenuation of 80
dB.
returns an IIR halfband interpolator, with additional properties specified by one or more
iirhalfbandinterp
= dsp.IIRHalfbandInterpolator(Name=Value
)Name-Value
pair arguments.
Example: iirhalfbandinterp = dsp.IIRHalfbandInterpolator(Specification="Filter
order and stopband attenuation")
creates an IIR halfband interpolator object
with filter order set to 9
and stopband attenuation set to
80
dB.
Properties
Unless otherwise indicated, properties are nontunable, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
release
function unlocks them.
If a property is tunable, you can change its value at any time.
For more information on changing property values, see System Design in MATLAB Using System Objects.
Specification
— Filter design parameters
"Transition width and stopband
attenuation"
(default) | "Filter order and stopband attenuation"
| "Filter order and transition width"
| "Coefficients"
Filter design parameters, specified as a character vector. When you set
Specification
to one of the filter design options, you can
specify the filter design parameters using the corresponding
FilterOrder
, StopbandAttenuation
, and
TransitionWidth
properties. Also, you can specify the design
method using DesignMethod
. When you set
Specification
to "Coefficients"
, you can
specify the coefficients directly.
FilterOrder
— Order of the IIR halfband filter
9 (default) | positive scalar integer
Order of the IIR halfband filter, specified as a positive scalar integer. If you set
DesignMethod
to "Elliptic"
, then
FilterOrder
must be an odd integer greater than one. If you set
DesignMethod
to "Quasi-linear phase"
, then
FilterOrder
must be a multiple of four.
Dependencies
To enable this property, set Specification
to
"Filter order and stopband attenuation"
or "Filter order
and transition width"
.
Data Types: single
| double
| int8
| int16
| int32
| int64
| uint8
| uint16
| uint32
| uint64
StopbandAttenuation
— Minimum attenuation needed in stopband
80 (default) | positive real scalar
Minimum attenuation needed in the stopband of the IIR halfband filter, specified as a positive real scalar. Units are in dB.
Dependencies
To enable this property, set Specification
to
"Filter order and stopband attenuation"
or "Transition
width and stopband attenuation"
.
Data Types: single
| double
TransitionWidth
— Transition width
4100 (default) | positive real scalar
Transition width of the IIR halfband filter, specified as a positive real scalar or in normalized frequency units (since R2023b).
If you set the
NormalizedFrequency
property to:
false
–– The value of the transition width is in Hz and must be less than half the output sample rate (2 ×SampleRate
property) value.true
–– The value of the transition width is in normalized frequency units. The value must be a positive scalar less than1.0
.When you set the
NormalizedFrequency
property totrue
while creating the object and you do not set the transition width, the default transition width is automatically set to normalized frequency units using the default sample rate at which the filter operates (2 × 44100) Hz. The filter in the halfband interpolator effectively runs at twice the sample rate of the input signal.When you set the
NormalizedFrequency
property totrue
after you create the object, the transition width must be specified in normalized units before you run the object algorithm. To specify the normalized frequency value, setNormalizedFrequency
totrue
and manually convert the frequency value in Hz to the normalized value using the input sample rate in Hz. For example, if the input sample rate is 22050 Hz, the corresponding transition width value in normalized units is TWHz/(2×Fs/2).iirhalfbandinterp = dsp.IIRHalfbandInterpolator; iirhalfbandinterp.NormalizedFrequency = true; iirhalfbandinterp.TransitionWidth = 4100/((2x22050)/2)
(since R2023b)
Dependencies
To enable this property, set Specification
to
"Transition width and stopband attenuation"
or "Filter
order and transition width"
.
Data Types: single
| double
DesignMethod
— Design method
"Elliptic"
(default) | "Quasi-linear phase"
Design method for the IIR halfband filter, specified as
"Elliptic"
or "Quasi-linear phase"
. When you set
this property to "Quasi-linear phase"
, the first branch of the
polyphase structure is a pure delay, which results in an approximately linear phase
response.
Dependencies
To enable this property, set Specification
to any accepted
value except "Coefficients"
.
NormalizedFrequency
— Flag to set frequencies in normalized units
false
(default) | true
Since R2023b
Flag to set frequencies in normalized units, specified as one of these values:
true
–– The transition width must be in the normalized frequency units and less than1.0
.When you set the
NormalizedFrequency
property totrue
while creating the object and you do not set the transition width, the default transition width is automatically set to normalized frequency units using the default sample rate at which the filter operates (2 × 44100) Hz. The filter in the halfband interpolator effectively runs at twice the sample rate of the input signal.When you set the
NormalizedFrequency
property totrue
after you create the object, the transition width must be specified in normalized units before you run the object algorithm. To specify the normalized frequency value, setNormalizedFrequency
totrue
and manually convert the frequency value in Hz to the normalized value using the input sample rate in Hz. For example, if the input sample rate is 22050 Hz, the corresponding transition width value in normalized units is TWHz/(2×Fs/2).iirhalfbandinterp = dsp.IIRHalfbandInterpolator; iirhalfbandinterp.NormalizedFrequency = true; iirhalfbandinterp.TransitionWidth = 4100/((2x22050)/2)
false
–– The transition width is in Hz. You can specify the input sample rate through theSampleRate
property.
Dependency
To enable this property, set Specification
to any accepted
value except "Coefficients"
.
Data Types: logical
SampleRate
— Input sample rate
44100 (default) | positive real scalar
Input sample rate, specified as a positive real scalar. Units are in Hz.
Dependency
To enable this property, set:
Specification
to any accepted value except"Coefficients"
.NormalizedFrequency
tofalse
. (since R2023b)
Data Types: single
| double
FilterBankInputPort
— Option to use object as synthesis filter bank
false
(default) | true
Option to use object as synthesis filter bank, specified as a logical value. If this
property is false
, dsp.IIRHalfbandInterpolator
acts as an interpolator. If this property is
true
, then dsp.IIRHalfbandInterpolator
acts as a synthesis filter bank and the
algorithm accepts two inputs: the lowpass and highpass subbands.
Dependencies
To enable this property, set Specification
to any accepted
value except "Coefficients"
.
Structure
— Filter structure used in coefficient mode
"Minimum multiplier"
(default) | "Wave Digital Filter"
Internal allpass filter implementation structure, specified as "Minimum
multiplier"
or "Wave Digital Filter"
. Each structure uses
a different coefficients set, independently stored in the corresponding object
property.
This property is not tunable.
Dependencies
To enable this property, set Specification
to
"Coefficients"
.
AllpassCoefficients1
— Allpass polynomial filter coefficients of first branch
[0.1284563; 0.7906755]
(default) | [0.1284563 0.1534; 0.7906755 0.6745]
Allpass polynomial filter coefficients of the first branch, specified as an
N-by-1
or
N-by-2
matrix. N is the
number of first-order or second-order allpass sections.
Tunable: Yes
Dependencies
To enable this property, set Specification
to
"Coefficients"
and Structure
to
"Minimum multiplier"
.
Data Types: single
| double
| int8
| int16
| int32
| int64
| uint8
| uint16
| uint32
| uint64
AllpassCoefficients2
— Allpass polynomial filter coefficients of second branch
[0.4295667]
(default) | [0.7906755 0.1534]
Allpass polynomial filter coefficients of the second branch, specified as an
N-by-1
or
N-by-2
matrix. N is the
number of first-order or second-order allpass sections.
Tunable: Yes
Dependencies
To enable this property, set Specification
to
"Coefficients"
and Structure
to
"Minimum multiplier"
.
Data Types: single
| double
| int8
| int16
| int32
| int64
| uint8
| uint16
| uint32
| uint64
WDFCoefficients1
— Allpass filter coefficients of first branch in Wave Digital Filter form
[0.1284563; 0.7906755]
(default) | [0.1284563 0.1534; 0.7906755 0.6745]
Allpass filter coefficients of the first branch in Wave Digital Filter form,
specified as an N-by-1
or
N-by-2
matrix. N is the
number of first-order or second-order allpass sections. All elements must have an
absolute value less than or equal to 1
.
This property is not tunable.
Dependencies
To enable this property, set Specification
to
"Coefficients"
and Structure
to
"Wave Digital Filter"
.
Data Types: single
| double
| int8
| int16
| int32
| int64
| uint8
| uint16
| uint32
| uint64
WDFCoefficients2
— Allpass filter coefficients of second branch in Wave Digital Filter form
[0.4295667]
(default) | [0.7906755 0.1534]
Allpass filter coefficients of the second branch in Wave Digital Filter form,
specified as an N-by-1
or
N-by-2
matrix. N is the
number of first-order or second-order allpass sections. All elements must have an
absolute value less than or equal to 1
.
This property is not tunable.
Dependencies
To enable this property, set Specification
to
"Coefficients"
and Structure
to
"Wave Digital Filter"
.
Data Types: single
| double
| int8
| int16
| int32
| int64
| uint8
| uint16
| uint32
| uint64
HasPureDelayBranch
— Make the first branch a pure delay
false
(default) | true
Flag to make the first allpass branch a delay, specified as a logical scalar. When
this property is true, the first branch is treated as a pure delay and the properties
AllpassCoefficients1
and WDFCoefficients1
do
not apply.
This property is not tunable.
Dependencies
To enable this property, set Specification
to
"Coefficients"
.
Delay
— Length of the delay
1
(default) | finite positive scalar
Length of the first branch delay, specified as a finite positive scalar. The value of this property specifies the number of samples by which you can delay the input to the first branch.
This property is not tunable.
Dependencies
To enable this property, set Specification
to
"Coefficients"
and HasPureDelayBranch
to
1.
Data Types: single
| double
HasTrailingFirstOrderSection
— Make the last section of the second branch as first order
false
(default) | true
Option to treat the last section of the second branch as first order, specified as a
logical scalar. When this property is 1 and the coefficients of the second branch are in
an N-by-2 matrix, the object ignores the second element of the last
row of the matrix. The last section of the second branch then becomes a first-order
section. When this property is set to 0
, the last section of the
second branch is a second-order section. When the coefficients of the second branch are
in an N-by-1 matrix, this property is ignored.
This property is not tunable.
Dependencies
To enable this property, set Specification
to
"Coefficients"
.
Usage
Description
implements a halfband synthesis filter bank for the inputs y
= iirhalfbandinterp(x1
,x2
)x1
and
x2
. x1
is the lowpass output of a halfband
analysis filter bank and x2
is the highpass output of a halfband
analysis filter bank. dsp.IIRHalfbandInterpolator
implements a synthesis
filter bank only when the FilterBankInputPort
property is
true
.
Input Arguments
x1
— Data input
column vector | matrix
Data input to the IIR halfband interpolator, specified as a column vector or a matrix. This signal is the lowpass output of a halfband analysis filter bank. If the input signal is a matrix, each column of the matrix is treated as an independent channel.
Data Types: single
| double
Complex Number Support: Yes
x2
— Second data input
column vector | matrix
Second data input to the synthesis filter bank, specified as a column vector or a matrix. This signal is the highpass output of a halfband analysis filter bank. If the input signal is a matrix, each column of the matrix is treated as an independent channel.
The size, data type, and complexity of both the inputs must be the same.
Data Types: single
| double
Complex Number Support: Yes
Output Arguments
y
— Output of interpolator
column vector | matrix
Output of the interpolator, returned as a column vector or a matrix. The number of rows in the interpolator output is twice the number of rows in the input signal.
Data Types: single
| double
Complex Number Support: Yes
Object Functions
To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named obj
, use
this syntax:
release(obj)
Specific to dsp.IIRHalfbandInterpolator
freqz | Frequency response of discrete-time filter System object |
freqzmr | Compute DTFT approximation of impulse response of multirate or single-rate filter |
filterAnalyzer | Analyze filters with Filter Analyzer app |
info | Information about filter System object |
cost | Estimate cost of implementing filter System object |
polyphase | Polyphase decomposition of multirate filter |
outputDelay | Determine output delay of single-rate or multirate filter |
Examples
Frequency response of Quasi-linear Phase IIR Halfband Interpolator
Create a minimum order lowpass IIR halfband interpolation filter. The filter has a transition width of 0.0930 in normalized frequency units, and a stopband attenuation of 80 dB.
IIRHalfbandInterp = dsp.IIRHalfbandInterpolator(... NormalizedFrequency=true,... DesignMethod="Quasi-linear phase");
Obtain filter coefficients
c = coeffs(IIRHalfbandInterp);
Plot the Magnitude and Phase response
freqz(IIRHalfbandInterp)
Design and Implement IIR Halfband Interpolator
Design an elliptic IIR halfband interpolator object of order 31 and a transition width of 0.1 using the designHalfbandIIR
function. Set the Verbose
argument to true
.
hbIIR = designHalfbandIIR(FilterOrder=31,TransitionWidth=0.1,DesignMethod="ellip",... Structure='interp',SystemObject=true,Verbose=true)
designHalfbandIIR(FilterOrder=31, TransitionWidth=0.1, DesignMethod="ellip", Structure="interp", Datatype="double", SystemObject=true, Passband="lowpass")
hbIIR = dsp.IIRHalfbandInterpolator with properties: Specification: 'Coefficients' FilterBankInputPort: false Structure: 'Minimum multiplier' HasPureDelayBranch: false AllpassCoefficients1: [8x1 double] AllpassCoefficients2: [7x1 double] HasTrailingFirstOrderSection: false
Create a dsp.DynamicFilterVisualizer
object and visualize the magnitude response of the filter.
dfv = dsp.DynamicFilterVisualizer(NormalizedFrequency=true); dfv(hbIIR);
The input is a cosine wave.
Fs = 1; Fc = 0.08; input = cos(2*pi*Fc*(0:39)'/Fs);
Interpolate the cosine signal using the IIR halfband interpolator.
output = hbIIR(input);
Plot the original and interpolated signals. In order to plot the two signals in the same plot, you must account for the output delay introduced by the IIR halfband interpolator and the scaling introduced by the filter. Use the outputDelay
function to compute the delay
introduced by the interpolator. Shift the output by this delay value.
Visualize the input and the resampled signals. The input and output values coincide every other sample, due to the interpolation factor of 2.
[delay,FsOut] = outputDelay(hbIIR,FsIn=Fs,Fc=Fc)
delay = 3.5090
FsOut = 2
nInput = (0:length(input)-1); tOutput = (0:length(output)-1)/FsOut-delay; stem(tOutput,output,'filled',MarkerSize=4); hold on; stem(nInput,input); hold off; xlim([-5,20]) legend('Interpolated by 2','Input signal','Location','best');
Extract Low Frequency Subband from Speech
Use a halfband analysis filter bank and interpolation filter to extract the low frequency subband from a speech signal.
Note: The audioDeviceWriter
System object™ is not supported in MATLAB Online.
Set up the audio file reader, the analysis filter bank, the audio device writer, and the interpolation filter. The sampling rate of the audio data is 22050 Hz. The halfband filter has an order of 21 and a transition width of 2 kHz.
afr = dsp.AudioFileReader('speech_dft.mp3',SamplesPerFrame=1024); filterspec = "Filter order and transition width"; Order = 21; TW = 2000; IIRHalfbandDecim = dsp.IIRHalfbandDecimator(... Specification=filterspec,FilterOrder=Order,... TransitionWidth=TW,SampleRate=afr.SampleRate); IIRHalfbandInterp = dsp.IIRHalfbandInterpolator(... Specification=filterspec,FilterOrder=Order,... TransitionWidth=TW,SampleRate=afr.SampleRate/2); ap = audioDeviceWriter(SampleRate=afr.SampleRate);
View the magnitude response of the halfband filter.
filterAnalyzer(IIRHalfbandDecim)
Read the speech signal from the audio file in frames of 1024 samples. Filter the speech signal into lowpass and highpass subbands with a halfband frequency of 5512.5 Hz. Reconstruct a lowpass approximation of the speech signal by interpolating the lowpass subband. Play the filtered output.
while ~isDone(afr) audioframe = afr(); xlo = IIRHalfbandDecim(audioframe); ylow = IIRHalfbandInterp(xlo); ap(ylow); end
Wait until the audio file ends, and then close the input file and release the audio output resource.
release(afr); release(ap);
Two-Channel Filter Bank
Use a halfband decimator and interpolator to implement a two-channel filter bank. This example uses an audio file input and shows that the power spectrum of the filter bank output does not differ significantly from the input.
Note: The audioDeviceWriter
System object™ is not supported in MATLAB Online.
Set up the audio file reader and audio device writer. Construct the IIR halfband decimator and interpolator. Finally, set up the spectrum analyzer to display the power spectra of the filter-bank input and output.
AF = dsp.AudioFileReader('speech_dft.mp3',SamplesPerFrame=1024); AP = audioDeviceWriter(SampleRate=AF.SampleRate); filterspec = "Filter order and transition width"; Order = 51; TW = 2000; IIRHalfbandDecim = dsp.IIRHalfbandDecimator(... Specification=filterspec,FilterOrder=Order,... TransitionWidth=TW,SampleRate=AF.SampleRate); IIRHalfbandInterp = dsp.IIRHalfbandInterpolator(... Specification=filterspec,FilterOrder=Order,... TransitionWidth=TW,SampleRate=AF.SampleRate/2,... FilterBankInputPort=true); SpecAna = spectrumAnalyzer(SampleRate=AF.SampleRate,... PlotAsTwoSidedSpectrum=false,... ShowLegend=true,... ChannelNames={'Input signal','Filtered output signal'});
Read the audio 1024 samples at a time. Filter the input to obtain the lowpass and highpass subband signals decimated by a factor of two. This is the analysis filter bank. Use the halfband interpolator as the synthesis filter bank. Display the running power spectrum of the audio input and the output of the synthesis filter bank. Play the output.
while ~isDone(AF) audioInput = AF(); [xlo,xhigh] = IIRHalfbandDecim(audioInput); audioOutput = IIRHalfbandInterp(xlo,xhigh); spectrumInput = [audioInput audioOutput]; SpecAna(spectrumInput); AP(audioOutput); end release(AF); release(AP); release(SpecAna);
Upsample and Interpolate Multichannel Input with IIR Halfband Interpolator
Create a halfband interpolation filter. The filter order is 51 with a transition width of 0.0930 in normalized frequency units.
filterspec = "Filter order and transition width"; Order = 51; TW = 0.0930; iirhalfbandinterp = dsp.IIRHalfbandInterpolator(... NormalizedFrequency=true,... Specification=filterspec,... FilterOrder=Order,... TransitionWidth=TW);
Use the filter to upsample and interpolate a multichannel input.
x = randn(1024,4); y = iirhalfbandinterp(x);
Algorithms
Polyphase Implementation with Halfband Filters
When you filter your signal, the IIR halfband interpolator uses an efficient polyphase implementation for halfband filters. You can use a polyphase implementation to move the upsampling operation after filtering. This change enables you to filter at a lower sampling rate.
IIR halfband filters are generally modeled using two parallel allpass filter branches.
Elliptic Design
The allpass filters for elliptic IIR halfband filter are given as
Quasi-Linear Phase Design
A near-linear phase response for IIR halfband filters is achieved by making one of the branches a pure delay. In this design, the cost of the filter increases.
The allpass filters for quasi-linear phase IIR halfband filter are
where, k is the length of the delay.
where N is the order of the IIR halfband filter.
You can represent the upsampling-by-2 operation followed by the filtering operation using this figure.
Using the multirate noble identity for upsampling, you can move the upsampling operation after filtering. This enables you to filter at a lower rate.
To efficiently implement the halfband interpolator, this algorithm replaces the upsampling operator, delay block, and adder with a commutator switch. The commutator switch starts on the first branch and takes input samples from the two branches alternately, one sample at a time. This doubles the sampling rate of the input signal. This is shown in the following figure.
Synthesis Filter Bank
Transfer function of the complementary highpass filter branch of the synthesis filter bank is given by
You can represent the synthesis filter bank as in this diagram.
The IIR halfband interpolator implements the synthesis portion of a two-band filter bank to synthesize a signal from lowpass and highpass subbands.
For more information on filter banks, see Overview of Filter Banks.
To summarize, the IIR halfband interpolator:
Filters the input before upsampling
Acts as a synthesis filter bank
Has a nonlinear phase response and uses few coefficients with the elliptic design method
Has near-linear phase response at the cost of additional coefficients with the quasi-linear phase design method, where one of the branches is a pure delay
References
[1] Lang, M. Allpass Filter Design and Applications. IEEE Transactions on Signal Processing. Vol. 46, No. 9, Sept 1998, pp. 2505–2514.
[2] Harris, F.J. Multirate Signal Processing for Communication Systems. Prentice Hall. 2004, pp. 208–209.
[3] Regalia, Phillip A., Sanjit K. Mitra, and P. P. Vaidyanathan. "The Digital All-Pass Filter: A Versatile Signal Processing Building Block." Proceedings of the IEEE. Vol. 76, Number 1, 1988, pp. 19-37.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
See System Objects in MATLAB Code Generation (MATLAB Coder).
This object supports code generation for ARM® Cortex®-M and ARM Cortex-A processors.
Version History
Introduced in R2015bR2023b: Support for normalized frequencies
When you set the NormalizedFrequency
property to
true
, you must specify the transition width in normalized frequency
units (0 to 1).
When you set the NormalizedFrequency
property to
true
while creating the object and you do not set the transition width,
the default transition width is automatically set to normalized frequency units using the
default sample rate at which the filter operates (2 × 44100) Hz. The filter in the halfband
interpolator effectively runs at twice the sample rate of the input signal.
iirhalfbandinterp = dsp.IIRHalfbandInterpolator(NormalizedFrequency=true)
iirhalfbandinterp = dsp.IIRHalfbandInterpolator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.1859 StopbandAttenuation: 80 DesignMethod: 'Elliptic' FilterBankInputPort: false NormalizedFrequency: true
When you set the NormalizedFrequency
property to
true
after you create the object, the transition width must be
specified in normalized units before you run the object
algorithm.
iirhalfbandinterp = dsp.IIRHalfbandInterpolator
iirhalfbandinterp = dsp.IIRHalfbandInterpolator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 4100 StopbandAttenuation: 80 DesignMethod: 'Elliptic' FilterBankInputPort: false NormalizedFrequency: false SampleRate: 22050
NormalizedFrequency
to true
and manually convert
the frequency values in Hz to normalized values using the input sample rate in Hz. For
example, if the input sample rate is 22050 Hz, the corresponding values in normalized units
are computed using these equations.iirhalfbandinterp.NormalizedFrequency = true; iirhalfbandinterp.TransitionWidth = 4100/((2x22050)/2)
iirhalfbandinterp = dsp.IIRHalfbandInterpolator with properties: Specification: 'Transition width and stopband attenuation' TransitionWidth: 0.1859 StopbandAttenuation: 80 DesignMethod: 'Elliptic' FilterBankInputPort: false NormalizedFrequency: true
See Also
Functions
freqz
|freqzmr
|filterAnalyzer
|info
|cost
|polyphase
|outputDelay
|designHalfbandIIR
Objects
Blocks
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