AN2099 - PSoC 1, PSoC 3, PSoC 4, and PSoC 5LP - Single-Pole Infinite Impulse Response (IIR) Filters.pdf

AN2099
PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP - Single-Pole Infinite Impulse
Response (IIR) Filters
Author: David Van Ess, Praveen Sekar
Associated Project: Yes
®
Associated Part Family: All PSoC 1, PSoC 3 and PSoC 5LP parts
Software Version: PSoC Creator™, PSoC Designer™
Related Application Notes: None
AN2099 describes a topology for a single-pole infinite impulse response (IIR) filter. It includes equations and software to
implement this topology; the associated example projects give the user access to filter routines in either assembly or C.
Figure 1. Single-Pole Passive RC Filters
Introduction
V(t)hp
In the real world analog signals are noisy; one example
might be the output voltage of a thermistor. It is often
undesirable to display or use this noisy data. The best way
to remove or “clean up” the noise is to apply a filter to the
signal. Ideally, the filter removes the noise and keeps the
signal of interest. Filters exist in the analog domain that
can be used to reduce noise. However, this results in extra
cost and power consumption of an analog filter. That is
where digital filters come in. IIR filters can be used to
approximate many common analog filters.
This application note derives the transfer function of a first
order IIR Low Pass and High Pass Filter. Based on these
transfer functions, the C and ASM code for a Low Pass
Filter are derived. Three example projects are provided
with this application note to provide hands-on examples of
how the filters work.
Infinite Impulse Response (IIR) Filters
An IIR filter is a recursive filter; that is, the output is used
to calculate future values. Theoretically, an impulse
injected into the input continues to flow through the signal
loop. It takes infinite time for the effect of the impulse to
die down completely.
The single-pole passive RC filter shown in Figure 1 has
the characteristics of an IIR filter. If you give an impulse
input, it takes an infinite time for the capacitor voltage to
go completely to zero.
We will build the topology of the single-pole IIR digital filter
from the single-pole passive RC filter shown in Figure 1.
www.cypress.com
V(t)in
V(t)lp
(a)
In Figure 1, the low-pass filter output is available across
the capacitor and the high-pass filter output is available
across the resistor.
The output voltage, Vlp, at any instant is a function of the
current flowing into the capacitor which is a function of
Vin-Vlp. This represents a negative feedback.
To construct a negative feedback equivalent to the analog
RC filter in Figure 1, consider the following statements:

At any instant, Vin-Vlp is divided by R to give a current,
which is then integrated to give Vlp.

Vin-Vlp represents the high-pass filtered output.
Figure 2 shows the negative feedback topology equivalent
to that in Figure 1.
Figure 2. Negative Feedback Topology for Single-Pole RC
Filter
V(t)hp
V(t)in
Document No. 001-38007 Rev. *H
1
R
1
sC
V(t)lp
1
PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
Equations 1 and 2 define its operation:
Vhp
Vhp  Vin  Vlp
Vin
Equation 1
Vhp
Vlp 
Vlp
Equation 2
Vin
sCR
Combining Equations 1 and 2 produces the transfer
functions in Equations 3 and 4:
Vhp
Vin
Vlp
Vin

sCR
1  sCR
Equation 3

1
1  sCR
Equation 4
1
2CR
a (1  z 1 )
a  z 1 (1  a )
Equation 10

1
a  z (1  a )
Equation 11
Equation 5
1
It can be shown that the roll-off frequency f0 for the digital
filter represented by Equations 6 and 7 can be
approximated by Equation 12. (See Appendix A)
f0 
These are the standard transfer functions for high-pass
and low-pass filters. The roll-off frequency f0 is shown in
Equation 5:
f0 

fs
2a
Equation 12
Equation 12 is the same as Equation 5 with RC = a. The
roll-off frequency is dependent on the sample frequency fs,
but more importantly the attenuation value (a). Changing
the attenuation value easily changes the filter’s roll-off
frequency. If you increase the attenuation factor, you can
lower the filter cut-off frequency. However, this results in
an increased filter settling time.
Figure 4. Filter Cut-off Versus Attenuation Factor
The topology for a sampled system is constructed from the
topology in Figure 2 and is shown in Figure 3.
Figure 3 shows that the integrator is replaced with an
accumulator (summer), and the term CR is replaced by a
-1
scaling factor. The z box represents a register that stores
the previous value of V(n)lp.
Figure 3. IIR Topology for Sampled Single-Pole RC Filters
Filter Settling Time
Equations 6 and 7 define its operation:
Vhp  Vin  Vlp z 1
Vlp 
Vhp
a
 Vlp z 1
Equation 6
For the analog RC low-pass filter shown in Figure 1, the
output for a step input will be an exponential rise defined
by Equation 11.
Equation 7
-1
Settling time in a low-pass filter is the time taken by the
output to reach a certain percentage of a step input.
-1
where z represents a unit sample delay. Replacing z
with a unit sample delay gives the difference (see
Equations 8 and 9).
Equation 8
Vhp ( n) V in Vlp (n  1)
Vlp ( n) 
Vhp ( n )
a
 Vlp ( n  1)
Equation 9
Vout  Vin (1  e
t
RC
)
Equation 13
For Vin = 1 V, R = 1 k, and C = 10 uF, the filter settling
(Vout Vs time) for a unit step input is shown by the blue line
in Figure 5.
Combining Equations 6 and 7, we get the transfer
functions in Equations 10 and 11.
www.cypress.com
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
Figure 5. Analog LPF Unit Step Response
To calculate the number (n) of samples required for the
output to reach 99.9% of the input, you must substitute
s[n] = 0.999 in Equation 14 and compute n.
 1
0.999  1  1  
 8
n
or
n  ~ 52
It takes 52 samples for the filter to reach 99.9% of the
input voltage. If you filter an ADC output using this LPF
(a = 8) and if the ADC sample rate is 52 sps, it takes
approximately a second for the filter to reach 99.9% of the
ADC input voltage.
To find the time taken by the filter to reach 99.9% of the
input, we substitute, Vout = 0.999; Vin = 1; R = 1 k;
C = 10 uF in Equation 13.
0.999 = (1 – e
-100t
) or
0.7 seconds are necessary to reach 99.9% of the step
input. This time depends on the choice of R and C.
Similar to the analog low-pass filter output, the digital
low-pass filter output also takes a finite time to reach
within a certain percentage of its input. In this case, the
settling time is determined by the attenuation factor, a.
It can be shown (see Appendix B) that for a unit step input,
the amplitude of the nth sample output, s[n], is given by
Equation 14.
1
2n
) * input voltage.
Table 1 shows the settling time for 10, 12, and 16-bit
accuracies.
For 10-bit accuracy, the output should reach within 0.1%
of the input value; For 12-bit accuracy, the output should
reach within 0.025% of the input value, and for 16-bit
accuracy, the output should reach within 0.0015% of the
input value. For a 1-V input, the filter achieves 10-bit
accuracy at 999.03 mV, 12-bit accuracy at 999.75 mV,
and 16-bit accuracy at 999.985 mV.
Table 1. Filter Settling Times
n
Equation 14
Equation 14 is the digital equivalent of Equation 13 with
Vin = 1. Figure 6 plots s[n] (the unit step response) Vs n
(the number of samples) for an attenuation factor of 8. The
red line shows the input and the blue line shows the filter
output.
Figure 6. Digital LPF Unit Step Response
The settling time for the filter output to reach n-bit
accuracy is the time taken by the filter to settle to within
(1 
t = 0.7 s
 1
sn  1  1  
 a
The filter settling time is also specified in terms of number
of bits.
Attenuation
Factor, a
10-bit
Accuracy
12-bit
Accuracy
16-bit
Accuracy
1
0
0
0
2
10
12
16
4
25
29
39
8
52
63
84
16
108
129
173
32
219
262
350
64
441
527
706
128
885
1058
1417
256
1765
2120
2838
How to Implement an IIR LPF
To implement the low-pass and high-pass filters shown in
Figure 3, we can directly use Equations 8 and 9. The
steps are as follows:
www.cypress.com
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
1.
Using your sample frequency (fs) and the roll-off
frequency that you want to use (f0), find the necessary
divisor (a) using Equation 12.
2.
Subtract the old Vlp from Vin. This is the new Vhp.
3.
Divide Vhp by a.
4.
Add the value generated in step 3 to the old Vlp. This
is the new Vlp. Note that only Vlp needs to be saved
for the calculation of the next values.
Step 3 requires you to perform a division (involves floating
point arithmetic) by the attenuation factor (a), which can
take any real value greater than unity. If you choose the
attenuation factor carefully, you can perform the division
with just shifts and add. Let us limit the attenuation factor
to this set of values:
256 
 256 256 256
a
,
,
,.....,

1 
 255 254 253
Equation 15
For example, if the attenuation factor is 256/99, Equation 9
becomes:
Vlp (n)  Vlp (n  1)  (Vin ( n)  Vlp (n  1)) *
99
256
Equation 16
99
64
32
2
1
1 1
1
1




  

256 256 256 256 256 4 8 128 256
This can be easily done by a combination of shifts and
adds, as shown by the C code snippet (Code 1). The
variable ‘filt’ denotes the low-pass filter output and the
variable ‘input’ denotes the low-pass filter input.
Binary Weighted IIR Filters
If the value of the attenuation factor, a, is a power of 2, it is
much easier to implement the filter. All that is required is
one shift and one addition. If the attenuation factor is 2^i,
the filtering can be done very easily using the following
line of code.
filt = filt + (input-filt) >> i;
This executes much faster and takes less code space as
well. In most cases, digital filtering is done to reduce the
noise, where the requirement to implement the actual
cut-off frequency is not stringent. Binary-weighted IIR
filters can be used in those cases.
Single-pole IIR Filter versus Moving
Average FIR Filter
A moving average filter is also commonly used for
reducing noise in the digital output. A moving average filter
is implemented by simply taking the average of the last N
samples as defined by Equation 17. The higher the value
of N, the lower is the filter cut-off.
y[n] 
Equation 17
The pros and cons of the moving average filter and the
single-pole IIR filter are as follows.

A single-pole IIR filter is easy to implement and
requires a storage space of just one memory element
to store the past output. However, the settling time
increases as the attenuation factor increases and the
filter takes infinite samples to settle to the input value.

A moving average FIR filter is relatively complex to
implement and requires a storage space of N
elements, where N is the number of samples
averaged. The filter takes just N samples to settle to
the input value.

A moving average filter with N elements has a
frequency response roll-off comparable to an IIR filter
with attenuation factor, a = N/2. Or, in other words,
the roll-off of a 32-tap moving average filter can be
matched by an IIR filter with an attenuation factor of
just 16. Figure 7 shows the frequency response
performance of a 32-element moving average filter
and a single-pole IIR filter of attenuation factor, 16.
Code 1
filt = filt + ((input-filt) >> 2) + ((inputfilt) >> 3) + ((input-filt) >> 7) + ((inputfilt) >> 8);
Similarly, any attenuation factor in the set can be
implemented with just shifts and additions.
The negative side of limiting the attenuation factor in this
way is that you may not have enough choices when the
attenuation factor gets higher. For example, if you want an
attenuation factor of 145, you must use 128 or 256. But if
you want an attenuation factor of 6.5, you can choose a
value very close to it, 6.5 which is 256/39. But, this should
not be a serious problem because higher attenuation
factors are generally not desired due to their higher
settling times.
For most practical purposes, you might require an
attenuation factor in the range of 4 to 64 and can easily
match it to one of the available values.
www.cypress.com
x[n]  x[n  1]  x[n  2]  ....  x[ N  1]
N
However, if you compare time responses, you can
see that it takes 108 samples (Table 1) for the IIR
filter to settle to 99.9% of the input while the moving
average FIR takes just 32 samples to settle to 100%
of the input signal.
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
filt = filt + ((input-filt) >> 4);
k = (filt>>8) + ((filt & 0x00000080) >> 7);
return k;
Table 2. Single-Pole IIR Versus Moving Average FIR
Single-pole IIR
Defining
parameter
Firmware
implementation
Memory
storage
required
Settling time
Moving Average FIR
Attenuation
factor, a
Number of samples
averaged, N
Easy
Relatively complex
Single-pole IIR
Moving Average FIR
Infinite
N
}
In this code, the statement
filt = filt + ((input-filt) >> 4);
Figure 7. Frequency Responses of Moving Average and
Single-Pole IIR filters
performs the filtering with an attenuation factor of 16. The
attenuation factor can be easily changed to the required
value by replacing this line of code with a code similar to
Code 1 (code 1 implements an attenuation factor of 2.56)
The variable ‘input’ is the input to the low-pass filter. The
variable ‘filt’ is a long int variable that accumulates the
filter running sum.
To avoid any loss of precision due to right shifts, we left
shift the input variable. The input is left-shifted by 8 so that
we can perform a right shift of 8 (maximum right shift for
a = 256) without losing precision. For this reason, the
variable filter should at least be 8 bits wider than the input.
The variable ‘filter’ is declared a long int variable while the
‘input’ is declared an int variable. This left-shift by 8 is
compensated by performing a right-shift by 8 before
returning the LPF output. This is done by the following
statement.
k = (filt>>8) + ((filt & 0x00000080) >> 7);
While right-shifting, perform rounding-off instead of
truncation. The additive term ((filt & 0x00000080) >>
7) checks the Most Significant bit (bit 8) of the bits shifted
out and performs the round-off before shifting the 8 bits
out.
Associated Projects
This application note includes four projects:

IIR_Filter_PSoC3_5 - PSoC Creator workspace
containing two projects, one for PSoC 3 (IIR_PSoC3)
and one for PSoC5 (IIR_PSoC5)

AN2099_asm - PSoC designer project for PSoC 1 in
assembly

AN2099_C -PSoC designer project for PSoC 1 in C.
The PSoC Creator project, IIR_PSoC3, performs low-pass
filtering on the 20-bit delta sigma ADC output and displays
the filtered and unfiltered value on the LCD.
The filter routine takes the filter input as an argument and
returns the filtered output after low-pass filtering the input
data with an attenuation factor of 16.
int32 LowPassFilter(int32 input)
{
int32 k;
input <<= 8;
www.cypress.com
In PSoC 4 and PSoC 5LP, the filtering can be done much
faster due to the availability of a single-cycle multiply
instruction in ARM cortex M0 and M3. In PSoC 4 and
PSoC 5LP, the filter routine for a low-pass filter (a =
256/17) looks like this:
Code 3
PSoC 3 Filter Implementation
Code 2
PSoC 4 and PSoC 5LP Filter
Implementation
int32 LowPassFilter(int32 input)
{
int32 k;
input <<= 8;
filt = filt + (((input-filt) >> 8) * 17);
k = (filt>>8) + ((filt & 0x00000080) >> 7);
return k;
}
The third statement performs the filtering. As you can see,
the division by 256 is done by right-shifting by 8 and the
multiplication is performed directly (without shift and add).
It is very easy to change the attenuation factor to any
value in Equation 15 by directly changing the value 17 to
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
the required value. Note that the division is performed
before the multiplication so that the variable filt stays
within the int32 limit.
If you program the PSoC Creator project in a PSoC 5LP
chip, you can see that the filtered value is comparatively
more stable than the raw ADC output. Try changing the
attenuation factor (a), and notice the change in settling
time and stability of the filtered output.
The PSoC 4 project is designed to work on the CY8CKIT042. This kit does not have an LCD. Thus the data is
output via UART on P4.1 (Pin 9 of J10). Connect this pin
to Pin 9 of J11 (the 12-pin header next to the USB
connector).
Plug a USB cable between your computer and the
CY8CKIT-042. Next, open the Cypress Bridge Control
Panel. In the Connected I2C/SPI/RX8 Ports: dialog, you
should see a COM port that corresponds to the CY8CKIT042 (for example COM8). Click on this COM port.
Next, click the Tools menu option and select Protocol
Configuration. Configure it to match Figure 8.
Figure 8 : RX8 Protocol Configuration
After you have configured the protocol, go to the Chart
Menu and select Variable Settings. Configure it to match
Figure 9.
Figure 9: Variable settings for COM Port
www.cypress.com
Now go to back to the editor and type the following
command:
rx8 [h=aa] @2RawData @1RawData
@2FilData @1FilData @0FilData [t=55]
@0RawData
Hit the Repeat button and then move to the Chart tab. You
should now be able to see a graph of the data. You can
move the input voltage around to see how the filtered and
raw data change.
Filter Feedforward
Filter feedforward eliminates the filter for a fast changing
input. If the input changes from 0 V to 5 V, the filter takes
108 iterations (attenuation factor = 16) to reach 4.995 V.
This delay can be reduced by including a feedforward term
to Code 3. Code 4 modifies Code 3 to include the
feedforward term.
Code 4
int32 LowPassFilter(int32 input)
{
int32 k;
int32 feedforward = (int32)100 * 256;
input <<= 8;
if ((input > (filt + feedforward)) || (input <
(filt - feedforward)))
{
filt = input;
}
else
{
filt = filt + ((input - filt) >> 4);
}
k = (filt>>8) + ((filt & 0x00000080) >> 7);
return k;
}
In Code 4, the ‘if else’ structure ensures that the filter is
not applied when the input value exceeds the current filter
value by more than 100. The feedforward term is
multiplied by 256 to bring it in the same order as the input
(the input is multiplied by 256 inside the filter code). If
there is a sudden change in input, say, from 0 V to 5 V (0
counts to 2^19 counts), the filter will not be applied and the
filtered value will reach the input value instantaneously.
The input will be filtered only when the input change is <
100 counts. To ensure that noise is filtered, the
feedforward coefficient should be higher than the peak-topeak noise.
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
PSoC 1 Filter Implementation
The two PSoC 1 based projects provide both low-pass
and high-pass filter functions written in assembly. The
functions have been written in assembly in PSoC 1
because the PSoC 1 CPU is low MIPS compared to that of
PSoC 3 and PSoC 5LP. The functions can be found in the
file IIRFilters.asm in the PSoC Designer project. One
project has the main file written in assembly and the other
project has the main file in C.
Both projects are similar to the PSoC Creator project in
that they take the ADC input and show both the ADC
output and low-pass filtered output in an LCD.
IIR High-Pass Filter – PSoC 1
The
iSimpleHighPassFilter function
in
the
file
IIRFilters.asm implements a high-pass filter with an
attenuation factor of 256. Equation 18 defines the roll-off
frequency. An attenuation of 256 is selected because it
can easily be implemented with a byte shift.
fs
f
f0 
 s
2 256 1608
Equation 18
For a sample rate of 5 ksps, the roll-off frequency is
3.1 Hz.
The function takes a 16-bit signed input and, using the old
Vlp, calculates the next value. The new 16-bit Vhp is
returned. This function assumes that 16-bit data is input
through the X (MSByte) and A (LSByte) registers. The
output is a 16-bit high-pass value returned through the X
and A registers. Code 5 shows the function.
This function is not called in the project. Only the
iSimpleLowPassFilter function is called. But if needed, you
can call this function in your project the same way the
iSimpleLowPassFilter function is called.
Code 4
area bss(RAM)
iVlp:
BLK 3
;[iVlp]
= MSByte
;[iVlp + 1] = LSByte
;[iVlp + 2] = Residue
area text(ROM,REL)
export SimpleHighPassFilter
export _SimpleHighPassFilter
;;---------------------------------;; SimpleHighPassFilter:
;;
;; Take input and output new
;; highpassvalue
;; INPUTS: X,A Vin
;; OUTPUTS: X,A Vhp
;;---------------------------------SimpleHighPassFilter:
www.cypress.com
_SimpleHighPassFilter:
sub A,[iVlp+1]
swap A,X
sbb A,[iVlp]
;Vhp now in A,X
cmp A,128 ;test if Vhp is neg
swap A,X
if1: jc elseif1 ;(if neg)
add [iVlp+2],A
swap X,A
adc [iVlp+1],A
swap A,X
adc [iVlp],ffh ;extended neg sign
ret
elseif1:;(pos)
add [iVlp+2],A
swap X,A
adc [iVlp+1],A
swap A,X
adc [iVlp],0 ;extended pos sign
ret
endif1:
;-----------------------------------
An initialization function is required to set the initial value
for Vlp. Code 6 shows this function:
Code 5
;;---------------------------------;; SimpleHighPassInit:
;;
;; Initializes the Vlp value
;; INPUTS: X,A Init Value
;; OUTPUTS: None.
;;---------------------------------SimpleHighPassInit:
_SimpleHighPassInit:
mov [iVlp],X
mov [iVlp + 1],A
mov [iVlp + 2],0
ret
;;----------------------------------
IIR Low Pass Filter – PSoC 1
The iSimpleLowPassFilter function implements a low-pass
filter with attenuation factor 256/12. The cut-off frequency
is given by Equation 19. To change the attenuation, you
have to change the SHIFT ACCUM sequence in the
assembly code.
f0 
f s * 12
f
 s
2 256 134
Equation 19
Code 7 is the function that takes a 16-bit signed input and,
using the old Vlp, calculates the next value. The new 16-bit
Vlp is returned:
Code 6
macro
asr
rrc
rrc
endm
SHIFT
[TempReg]
[TempReg + 1]
[TempReg + 2]
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
macro
mov
add
mov
adc
mov
adc
endm
;;---------------------------------SimpleLowPassInit:
_SimpleLowPassInit:
mov [iVlp1],X
mov [iVlp1 + 1],A
mov [iVlp1 + 2],0
ret
;;----------------------------------
ACCUM
A,[TempReg + 2]
[iVlp1 + 2],A
A,[TempReg + 1]
[iVlp1 + 1],A
A,[TempReg]
[iVlp1],A
Summary
export iSimpleLowPassFilter
export _iSimpleLowPassFilter
;;---------------------------------;; iSimpleLowPassFilter:
;;
;; Take input and output new
;; higpassvalue
;; INPUTS: X,A Vin
;; OUTPUTS: X,A Vhp
;;---------------------------------iSimpleLowPassFilter:
_iSimpleLowPassFilter:
push A
mov A,0
sub A,[iVlp1+2]
mov [TempReg + 2],A
pop A
sbb A,[iVlp1+1]
mov [TempReg + 1],A
mov A,X
sbb A,[iVlp1]
mov [TempReg],A
Single-pole IIR filters are useful for removing noise from
useful signals. Four projects have been presented that can
easily be tailored to your filtering requirements.
SHIFT
SHIFT
SHIFT
SHIFT
SHIFT
ACCUM ;32
SHIFT
ACCUM ;64
mov
A,[iVlp1 + 2]
add
mov
mov
adc
swap
adc
A,128
X,[iVlp1]
A,[iVlp1 + 1]
A,0
X,A
A,0
swap X,A
ret
;-----------------------------------
This function also requires an initialization function. It is
shown in Code 7:
Code 7
export SimpleLowPassInit
export _SimpleLowPassInit
;;---------------------------------;; SimpleLowPassInit:
;;
;; Initializes the Vlp value
;; INPUTS: X,A Init Value
;; OUTPUTS: None.
www.cypress.com
Document No. 001-38007 Rev. *H
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PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
Appendix A
The actual cut-off frequency for the single-pole low-pass
filter can be derived from Equation 20
Vlp
Vin

1
a  z 1 (1  a )
-1
Equation 20
-jw
Substituting z = e , we get,
Vlp
Vin
Vlp
Vin


Equation 21 gives the actual expression for the cut-off
frequency for the single-pole low-pass IIR filter. However,
for a >8 (filter attenuation factors commonly used are
between 8 and 64), Equation 21 can be approximated by
Equation 23:
f0 
1
ae
 jw
(1  a )
Vin

1
a  (cos w  j sin w)(1  a )
Table 3. Cut-off Frequency Approximation Error
1
(a  (1  a) cos w)
2
 ((1  a ) sin w)
2

Equation 21
Vlp
Vin

(1  2a
1
2
 2a (1  (1  a ) cos w
At the cut-off frequency (w = w0),
1

2
(1  2a
Vlp
Vin
=

1
2
1
2
 2a (1  (1  a ) cos w0
Equation 23
Table 3 shows the difference between the two cut-off
frequencies (Equations 12 and 21) at different values of
attenuation factor at a sample frequency of 10 kHz.
The magnitude of the transfer function is given by
Equation 21
Vlp
fs
2a
a
Cut-off Calculated
using Equation 12
Actual Cut-off
(Equation 21)
2
795.7747155
1150.267281
4
397.8873577
461.0511704
8
198.9436789
212.8383013
16
99.47183943
102.7519182
32
49.73591972
50.53386767
64
24.86795986
25.0648066
128
12.43397993
12.48286862
256
6.216989965
6.229172189

or
f0 


fs
1

cos1 1 
2
 2a(a  1) 
Equation 22
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Document No. 001-38007 Rev. *H
9
PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
Appendix B
The settling time of the single-pole digital IIR filter can be
derived from Equation 11, which is re-written below
Vlp
Vin

1
a  z (a  1)
 1  1 
hn   1  
 a  a 
n
Equation 26
1
Equation 24
If you rearrange Equation 24, you get Equation 25, which
is the frequency response of a digital low-pass filter with
attenuation factor, a.
1
1
H z    
 a  1  z 1 1  1 


 a
1 n  1
sn    1  
a 0  a
n
Equation 27
Simplifying Equation 27, we get Equation 28
Equation 25
If you take inverse z-transform, you get the impulse
response of the filter (h[n]).
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Summation of the impulse response over 0 to n samples
gives the unit step response (s[n]).
 1
sn  1  1  
 a
Document No. 001-38007 Rev. *H
n
Equation 28
10
PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
Document History
®
Document Title: PSoC 1, PSoC 3, PSoC 4, and PSoC 5LP - Single-Pole Infinite Impulse Response (IIR) Filters – AN2099
Document Number: 001-38007
Revision
ECN
Orig. of
Change
Description of Change
Submission
Date
**
1520284
DWV
10/01/2007
Old application note updated to CY web.
*A
2711571
YARA
05/27/2009
Added CY8C29x66, CY8C27x43, CY8C24x23A, CY8C24x94, CY8C21xxx, and
CY8C20xxx part families.
Updated Software version to PD 5.0.
*B
3248285
DSG
05/04/2011
Added section Implementation in C
*C
3394927
PFZ
10/12/2011
Major rewrite of the document
*D
3457966
PFZ
12/09/2011
Template Update
Updated project for PSoC Creator 2.0
*E
3492061
PFZ
01/11/2012
Fixed Table 2.
*F
3806325
RRSH
11/27/2012
Updated for PSoC 5LP.
*G
4202704
TDU
11/26/2013
Added PSoC 4 Project.
Fixed Equation 14.
Corrected equation reference in Appendix B.
*H
4373327
TDU
05/08/2014
No Change
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Document No. 001-38007 Rev. *H
11
PSoC® 1, PSoC 3, PSoC 4, and PSoC 5LP – Single-Pole Infinite Impulse Response (IIR) Filters
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Document No. 001-38007 Rev. *H
12
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