## Posted tagged ‘saturation’

### Implementation of FIR Filtering in C (Part 2)

October 8, 2009

In Part 1 I showed how to code a FIR filter in C using floating point. In this lesson I will show how to do the same thing using fixed point operations. The code example below will demonstrate the application of fixed point multiplication, rounding and saturation. The code has definitions for the FIR filtering function, followed by an example test program.

firFixed

And here is the code example:

```
#include <stdio.h>
#include <stdint.h>

//////////////////////////////////////////////////////////////
//  Filter Code Definitions
//////////////////////////////////////////////////////////////

// maximum number of inputs that can be handled
// in one function call
#define MAX_INPUT_LEN   80
// maximum length of filter than can be handled
#define MAX_FLT_LEN     63
// buffer to hold all of the input samples
#define BUFFER_LEN      (MAX_FLT_LEN - 1 + MAX_INPUT_LEN)

// array to hold input samples
int16_t insamp[ BUFFER_LEN ];

// FIR init
void firFixedInit( void )
{
memset( insamp, 0, sizeof( insamp ) );
}

// the FIR filter function
void firFixed( int16_t *coeffs, int16_t *input, int16_t *output,
int length, int filterLength )
{
int32_t acc;     // accumulator for MACs
int16_t *coeffp; // pointer to coefficients
int16_t *inputp; // pointer to input samples
int n;
int k;

// put the new samples at the high end of the buffer
memcpy( &insamp[filterLength - 1], input,
length * sizeof(int16_t) );

// apply the filter to each input sample
for ( n = 0; n < length; n++ ) {
// calculate output n
coeffp = coeffs;
inputp = &insamp[filterLength - 1 + n];
acc = 1 << 14;
// perform the multiply-accumulate
for ( k = 0; k < filterLength; k++ ) {
acc += (int32_t)(*coeffp++) * (int32_t)(*inputp--);
}
// saturate the result
if ( acc > 0x3fffffff ) {
acc = 0x3fffffff;
} else if ( acc < -0x40000000 ) {
acc = -0x40000000;
}
// convert from Q30 to Q15
output[n] = (int16_t)(acc >> 15);
}

// shift input samples back in time for next time
memmove( &insamp[0], &insamp[length],
(filterLength - 1) * sizeof(int16_t) );

}

//////////////////////////////////////////////////////////////
//  Test program
//////////////////////////////////////////////////////////////

// bandpass filter centred around 1000 Hz
// sampling rate = 8000 Hz
// gain at 1000 Hz is about 1.13

#define FILTER_LEN  63
int16_t coeffs[ FILTER_LEN ] =
{
-1468, 1058,   594,   287,    186,  284,   485,   613,
495,   90,  -435,  -762,   -615,   21,   821,  1269,
982,    9, -1132, -1721,  -1296,    1,  1445,  2136,
1570,    0, -1666, -2413,  -1735,   -2,  1770,  2512,
1770,   -2, -1735, -2413,  -1666,    0,  1570,  2136,
1445,    1, -1296, -1721,  -1132,    9,   982,  1269,
821,   21,  -615,  -762,   -435,   90,   495,   613,
485,  284,   186,   287,    594, 1058, -1468
};

// number of samples to read per loop
#define SAMPLES   80

int main( void )
{
int size;
int16_t input[SAMPLES];
int16_t output[SAMPLES];
FILE   *in_fid;
FILE   *out_fid;

// open the input waveform file
in_fid = fopen( "input.pcm", "rb" );
if ( in_fid == 0 ) {
printf("couldn't open input.pcm");
return;
}

// open the output waveform file
out_fid = fopen( "outputFixed.pcm", "wb" );
if ( out_fid == 0 ) {
printf("couldn't open outputFixed.pcm");
return;
}

// initialize the filter
firFixedInit();

// process all of the samples
do {
size = fread( input, sizeof(int16_t), SAMPLES, in_fid );
// perform the filtering
firFixed( coeffs, input, output, size, FILTER_LEN );
// write samples to file
fwrite( output, sizeof(int16_t), size, out_fid );
} while ( size != 0 );

fclose( in_fid );
fclose( out_fid );

return 0;
}

```

The first thing to notice is that the definitions for the input sample storage and handling are nearly the same as for the code in Part 1. The only difference is that the storage type is a 16 bit integer instead of double precision floating point.

The next difference is the inclusion of rounding in the calculation of each output. Rounding is used when converting the calculated result from a Q30 format number to Q15. Notice that I have loaded the rounding constant into the accumulator value (acc) at the beginning of the loop rather than adding it at the end. This is a small optimization commonly seen in code for FIR filters. If you are coding in assembly language, and your chip has a rounding instruction, it may be better to do the rounding at the end (depending on what the instruction actually does).

The multiplication itself is now an integer multiplication of two 16 bit values, each of which is a Q15 number. The accumulator variable is 32 bits, and holds a Q1.30 format number. There is one bit for the sign, one integer bit, and thirty fractional bits. Notice that I have cast each multiplier to a 32 bit value. Failure to do so should result in a 16 bit product and produce wrong results.

Next comes the overflow handling for converting from Q1.30 to Q30. The code checks for values beyond the limits of the largest/smallest Q30 number (no integer bits), and saturates if necessary.

Finally, a right shift by 15 is used to convert the MAC result from Q30 to Q15 and the result is stored to the output array.

The test program is simpler than the one in Part 1 because there is no need to convert the input and output samples to or from floating point. The most important thing to note is the change in the filter coefficient array. To generate these coefficients, I took the floating point coefficients from Part 1, multiplied by 32768, and then rounded to the nearest integer. The coefficients are in Q15 format, and note that none of the original floating point coefficients are close to one. Multiplying by 32768 would cause a problem for any coefficients larger than 32767/32768 or less than -1.

As in Part 1, the test input file should be 16 bit samples at a sampling rate of 8000 Hz. Try using a 1000 Hz tone with noise added to it. The bandpass filter should remove a large portion of the noise.

The filter has greater than unity gain at 1000 Hz and the gain is about 1.13. Admittedly it is not a great filter design, but it tests the saturation code if a full scale 1000 Hz sine wave is used as an input. Try a full scale 1000 Hz tone input with and without the saturation check and see what the difference is. There should be a small amount of distortion with the saturation code present (due to slight flattening of tone) and a large amount of distortion without it (due to overflow).

A fixed point FIR filter is fairly easy to implement. Personally I find the sample storage and movement to be the most difficult part. In Part 3 of this lesson I will demonstrate how to run multiple FIR filters on the same input.

### Overflow Handling in Fixed Point Computations

August 10, 2009

Overflow handling is an important consideration when implementing signal processing algorithms. If overflow is not controlled appropriately it can lead to problems such as detection errors, or poor quality audio output. Typical digital signal processing CPUs include hardware support for handling overflow. Some RISC processors may include these modes as well. (In fact I helped define and implement such modes for the 32 bit MIPS processor core used in many Broadcom products). These processors often have a “saturating” mode that sets an instruction result to a minimum or maximum value on an overflow condition. (The term “saturating” comes from analog electronics, in which an amplifier output will be limited, or clipped, between fixed values when a large input is applied.) Commonly the CPU will limit the result to a 32 bit twos complement integer (0x7FFFFFFF or 0x80000000). For unsigned operations, the result would be limited to 0xFFFFFFFF. There are a number of situations in which overflow can occur, and I will discuss some of them below.

Overflow with twos complement integers occurs when the result of an addition or subtraction is larger the largest integer that can be represented, or smaller than the smallest integer. In fixed point representation, the largest or smallest value depends on the format of the number. I will assume Q31 in a 32 bit register for any examples that follow. In this case, a CPU with saturation arithmetic would set the result to -1 or (just below) +1 on an overflow, corresponding to the integer values 0x80000000 and 0x7FFFFFFF.

Overflow in addition can only occur when the sign of the two numbers being added is the same. Overflow in subtraction can occur only when a negative number is subtracted from a positive number, or when a positive number is subtracted from a negative number.

#### Negation

There is one case where negation of a number causes an overflow condition. When the smallest negative number is negated, there is no way to represent the corresponding positive value in twos complement. For example, the value -1 in Q31 is 0x80000000. When this number is negated (flip the bits and add one) the result is again -1. If the saturation mode is set, then the CPU will set the result to 0x7FFFFFFF (just less than +1).

#### Arithmetic Shift

Overflow can occur when shifting a number left by 1 to n bits. In fixed point computations, left shifting is used to multiply a fixed point value by a power of two, or to change the format of a number (Q15 to Q31 for example). Again, many CPUs have saturation modes to set the output to the minimum or maximum 32 bit integer (depending on whether the original number was positive or negative). Furthermore, a common feature is an instruction that counts the number of leading ones or zeros in a number. This helps the programmer avoid overflow since the number of leading sign bits determines how large a shift can be done without causing overflow.

Overflow will not occur when right shifting a number.

#### Multiplication

Overflow doesn’t really occur during multiplication if the result register has enough bits (32 bits if two 16 bit numbers are multiplied). But it is partly a matter of interpretation. When multiplying a fixed point value of -1 by -1 (0x8000 by 0x8000 using Q15 numbers), the result is +1. If the result is interpreted as a Q1.30 number (one integer bit and 30 fractional bits) then there is no problem. If the result is to be a Q30 number (no integer bits) then an overflow condition has occurred. And if the number was to be converted to Q31 (by shifting the result left by 1) then an overflow would occur during the left shift. The overall affect would be that -1 times -1 equals -1.

I have used a CPU that handles this special case with saturation hardware. Some CPUs have a multiplication mode that shifts the product left by one bit after a multiply operation. The reason for doing so is to create a Q31 result when two Q15 numbers are multiplied. Then if a Q15 result is desired, it can be found by storing the upper 16 bits of the result register (if the register is only 32 bits). The saturating mode automatically sets the result to 0x7FFFFFFF when the number 0x8000 is multiplied by itself, and the “shift left by one” multiplication mode is enabled.

A very often used operation in DSP algorithms is the “multiply accumulate” or “MAC”, where a series of numbers is multiplied and added to a running sum. I would recommend not using the “left shift by one” mode if possible when doing MACs, since this only increases the chance for overflow. A better technique is to keep the result as Q1.30, and then handle overflow if converting the final result to Q31 or Q15 (or whatever). This is also a good technique to use on CPUs without saturation modes, since the number of overflow checks can be greatly reduced in some cases.

#### Division

Overflow in division can occur when the result would have more bits than was calculated. For example, if the magnitude of the numerator is several times larger than that of the denominator, than the result must have enough bits to represent numbers larger than one. Overflow can be avoided by carefully considering the range of numbers being operated on, and calculating enough bits for the result. I have not seen a CPU that implements a saturation mode for division.

Division by 0 is undefined, and not really an overflow case.

#### Conclusion

Many CPUs include hardware supported handling of overflow using saturation modes. These modes are useful, but it is better to avoid overflow in the first place if possible. This can lead to more accurate results in computations. And when using a CPU without saturation arithmetic, it is best to design the arithmetic operations so that the number of overflow checks is minimized.