Simple Low Pass IIR Filter Implementation Using the C Language

Simple Low Pass IIR Filter Implementation Using the C Language

1. Abbreviation

IIR – Infinite Impulse Response Filter
RISC – Reduced Instruction Set Computing

2. Introduction

The linear analog circuit is described with the linear differential equation or with the Laplace transformation (s plane). For example: the RC circuit (low pass filter) is described with the first order differential equation:

or in the Laplace form:

The analog method experience can be used for the digital technical. The linear digital methods are used the difference equation and z transformation (z plane).

There are few transformation methods of the s-plane (analog space) to z-plane (digital space): backward Euler, impulse invariance, bilinear. The article is described the simple first order filter. Backward Euler method is suitable for the case.

Insert the (5) in the (2)

Specify

Then the z-transformation

And the difference equation

3. First Order Digital IIR Filter

The low pass IIR filter first order using the backward Euler (or impulse invariance) transformation is described with the (8) and (9).

Example of the A value calculation
Sampling fs=20kHz
Filter passband fb=1kHz

Then
RC=1/(2πfb)=1/(2*3.14*1000Hz) ≈ 1.6e-4s
Ts=1/fs=1/20000Hz=0.5e-4s
A=Ts/(Ts+RC)=0.5e-4/(0.5e-4+1,6e-4) ≈ 0.24
y(n)=y(n-1)+0.24*(x(n)-y(n-1))

Notes
1. The value A is less as 1 => A < 1
2. The filter can be interpreted as weighting filter: y(n)=(1-A)y(n-1)+Ax(n). The input x(n) signal will be taken with wight A and previous history y(n-1) with the weight (1-A). If the input x(n) signal is not trusted then the value A shall be taken lower value.
3. The filter can be realized only with the one multiplication: y(n)=y(n-1)+A(x(n)-y(n-1)
4. IIR filter realization with fixed point arithmetic: output history y(n-1) shall be saved with the double accuracy

3.1 First Order Digital IIR Filter. Fixed Point Realization with the One Multiplication.

Fixed point realization shall be scaled. For example: input x and output y will be used signed 16 bits, scaling S=2^16 =0x10000=65536 and history Sy(n-1) will be saved signed 32 bits

Sy(n)=Sy(n-1)+SA(x(n)-y(n-1))   (10)

Output value without scaling:

y(n) = Sy(n-1)+SA(x(n)-y(n-1) +/- 0.5*S)/S   (11) where 0.5S is round value

C-Code:

sint16 IIR_Filter_First_Order(sint16 x_input)
{
  sint32 sy_round;
  sy_round = ROUND_MACRO(sy_history);
  sy_history = sy_history + SA_VALUE*((sint32)(x_input – (sy_round>>SCALING_SHIFT)));
  sy_round = ROUND_MACRO(sy_history);
  return ((sint16)(sy_round>>SCALING_SHIFT));
}

3.2 First Order Digital IIR Filter. Fixed Point Realization with the Shift Operation.

Instead of the multiplication can be used the arithmetic shift operation (multiplication/division on the 2^k). The A value shall be taken 2^-k. Then the (9) can be written

Using the scaling S=2^k in the (10)

The method can be used for example for the RISC microprocessor which has not the multiplication instruction in the assembler.

C-Code:

sint16 IIR_Filter_First_Order(sint16 x_input)
{
  sint32 sy_round;
  sy_round = ROUND_MACRO(sy_history);
  sy_history = sy_history + ((sint32)(x_input – (sy_round>>K_VALUE)));
  sy_round = ROUND_MACRO(sy_history);
  return ((sint16)(sy_round>>K_VALUE));
}

3.3 First Order Digital IIR Filter. Float Arithmetic.

The equation (9) shall be used in the case.

C-Code:

sint16 IIR_Filter_First_Order(sint16 x_input)
{
  sint16 y_round;
  y_history = y_history + A_VALUE*(((float)x_input – y_history));
  y_round = ROUND_MACRO(y_history);
  return y_round;
}

4. Appendix. Solution of the differential and difference first order equations

Consider the (1) differential equation with the input step signal:

Then the solution of the differential equation:

Consider the (9) difference equation with the input step signal:

Then the solution of the difference equation:

5. Appendix. C-Source Codes

You can use the follow C-Source codes for your applications:

IIRfirstOrderFixedPoint.c/h – implementation with single multiplication and fixed point arithmetic

IIRfirstOrderFloat.c/h – implementation with single multiplication and float point arithmetic

IIRfirstOrderShift.c/h – implementation with shift operation instead of multiplication and fixed point arithmetic

6. Literature / References

[1] Harry Y.-F. Lam „Analog and digital filters: design and realization“, Prentice-Hall Inc, 1979

[2] Lawrence R. Rabiner, Bernard Gold „Theory and application of digital signal processing“, Prentice-Hall Inc, 1975