Implementation of FIR Filtering in C (Part 1)

Posted October 1, 2009 by Shawn
Categories: FIR filters

Tags: , ,

In this lesson I will show how to code a finite impulse response (FIR) digital filter in the C programming language using floating point operations. Understanding the basic FIR filter is a stepping stone to understanding more complicated filters such as polyphase FIR filters and adaptive FIR filters.

A FIR filter is one in which the output depends only on the inputs (and not on previous outputs). The process is a discrete-time convolution between the input signal and the impulse response of the filter to apply. I will call this impulse response “the filter coefficients.” The following equivalent equations describe the convolution process:

equation 1: y(n)=\sum_{k=0}^{N-1}h(k)x(n-k)

equation 2: y(n)=\sum_{k=n-(N-1)}^{n}h(n-k)x(k)

where h is the impulse response, x is the input, y[n] is the output at time n and N is the length of the impulse response (the number of filter coefficients). For each output calculated, there are N multiply-accumulate operations (MACs). There is a choice of going through the filter coefficients forward and the input backwards (equation 1), or through the filter coefficients backwards and the input forwards (equation 2). In the example that follows, I will use the form in equation 1.

In my code example, samples will be stored in a working array, with lower indices being further back in time (at the “low” end). The input samples will be arranged in an array long enough to apply all of the filter coefficients. The length of the array required to process one input sample is N (the number of coefficients in the filter). The length required for M input samples is N – 1 + M.

A function will be defined that filters several input samples at a time. For example, a realtime system for processing telephony quality speech might process eighty samples at a time. This is done to improve the efficiency of the processing, since the overhead of calling the function is eighty times smaller than if it were called for every single sample. The drawback is that additional delay is introduced in the output.

The basic steps for applying a FIR filter are the following:

  1. Arrange the new samples at the high end of the input sample buffer.

  2. Loop through an outer loop that produces each output sample.

  3. Loop through an inner loop that multiplies each filter coefficient by an input sample and adds to a running sum.

  4. Shift the previous input samples back in time by the number of new samples that were just processed.

The code example defines an initialization function and a filtering function, and includes a test program. Download the PDF from the following link to see the code, or view it in-line below:

firFloat

#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
double insamp[ BUFFER_LEN ];

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

// the FIR filter function
void firFloat( double *coeffs, double *input, double *output,
       int length, int filterLength )
{
    double acc;     // accumulator for MACs
    double *coeffp; // pointer to coefficients
    double *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(double) );

    // apply the filter to each input sample
    for ( n = 0; n < length; n++ ) {
        // calculate output n
        coeffp = coeffs;
        inputp = &insamp[filterLength - 1 + n];
        acc = 0;
        for ( k = 0; k < filterLength; k++ ) {
            acc += (*coeffp++) * (*inputp--);
        }
        output[n] = acc;
    }
    // shift input samples back in time for next time
    memmove( &insamp[0], &insamp[length],
            (filterLength - 1) * sizeof(double) );

}

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

// bandpass filter centred around 1000 Hz
// sampling rate = 8000 Hz

#define FILTER_LEN  63
double coeffs[ FILTER_LEN ] =
{
  -0.0448093,  0.0322875,   0.0181163,   0.0087615,   0.0056797,
   0.0086685,  0.0148049,   0.0187190,   0.0151019,   0.0027594,
  -0.0132676, -0.0232561,  -0.0187804,   0.0006382,   0.0250536,
   0.0387214,  0.0299817,   0.0002609,  -0.0345546,  -0.0525282,
  -0.0395620,  0.0000246,   0.0440998,   0.0651867,   0.0479110,
   0.0000135, -0.0508558,  -0.0736313,  -0.0529380,  -0.0000709,
   0.0540186,  0.0766746,   0.0540186,  -0.0000709,  -0.0529380,
  -0.0736313, -0.0508558,   0.0000135,   0.0479110,   0.0651867,
   0.0440998,  0.0000246,  -0.0395620,  -0.0525282,  -0.0345546,
   0.0002609,  0.0299817,   0.0387214,   0.0250536,   0.0006382,
  -0.0187804, -0.0232561,  -0.0132676,   0.0027594,   0.0151019,
   0.0187190,  0.0148049,   0.0086685,   0.0056797,   0.0087615,
   0.0181163,  0.0322875,  -0.0448093
};

void intToFloat( int16_t *input, double *output, int length )
{
    int i;

    for ( i = 0; i < length; i++ ) {
        output[i] = (double)input[i];
    }
}

void floatToInt( double *input, int16_t *output, int length )
{
    int i;

    for ( i = 0; i  32767.0 ) {
            input[i] = 32767.0;
        } else if ( input[i] < -32768.0 ) {
            input[i] = -32768.0;
        }
        // convert
        output[i] = (int16_t)input[i];
    }
}

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

int main( void )
{
    int size;
    int16_t input[SAMPLES];
    int16_t output[SAMPLES];
    double floatInput[SAMPLES];
    double floatOutput[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( "outputFloat.pcm", "wb" );
    if ( out_fid == 0 ) {
        printf("couldn't open outputFloat.pcm");
        return;
    }

    // initialize the filter
    firFloatInit();

    // process all of the samples
    do {
        // read samples from file
        size = fread( input, sizeof(int16_t), SAMPLES, in_fid );
        // convert to doubles
        intToFloat( input, floatInput, size );
        // perform the filtering
        firFloat( coeffs, floatInput, floatOutput, size,
               FILTER_LEN );
        // convert to ints
        floatToInt( floatOutput, output, size );
        // write samples to file
        fwrite( output, sizeof(int16_t), size, out_fid );
    } while ( size != 0 );

    fclose( in_fid );
    fclose( out_fid );

    return 0;
}

First is the definition of the working array to hold the input samples. I have defined the array length to handle up to 80 input samples per function call, and a filter length up to 63 coefficients. In other words, the filter function that uses this working array will support between 1 to 80 input samples, and a filter length between 1 and 63.

Next is the initialization function, which simply zeroes the entire input sample array. This insures that all of the inputs that come before the first actual input are set to zero.

Then comes the FIR filtering function itself. The function takes pointers to the filter coefficients, the new input samples, and the output sample array. The “length” argument specifies the number of input samples to process (must be between 0 and 80 inclusive). The “filterLength” argument specifies the number of coefficients in the filter (must be between 1 and 63 inclusive). Note that the same filter coefficients and filterLength should be passed in each time the firFloat function is called, until the entire input is processed.

The first step in the function is the storage of the new samples. Note closely where the samples are placed in the array.

Next comes the outer loop that processes each input sample to produce one output sample. The outer loop sets up pointers to the coefficients and the input samples, and zeroes an accumulator value for the multiply-accumulate operation in the inner loop. It also stores each calculated output. The inner loop simply performs the multiply-accumulate operation.

Finally comes the step to move the input samples back in time. This is probably the most difficult step in the FIR filter and the part that always takes me the longest to work out. Pay close attention to the position of the samples to move, and the number of samples moved. The amount of time shift is equal to the number of input samples just processed. The number of samples to move is one less than the number of coefficients in the filter. Also note the use of the memmove function rather than the memcpy function, since there is a potential for moving overlapping data (whether or not the data overlaps depends on the relative lengths of the input samples and number of coefficients).

That concludes the code example for the FIR filter. In Part 2, I will show how to do the same thing using fixed point operations.

Fixed Point Extensions to the C Programming Language

Posted September 10, 2009 by Shawn
Categories: dsp, fixed point

Tags: , , ,

Recently I ran across an ISO specification for extensions to the C programming language to support fixed point types. The types are defined in a header file called stdfix.h. I have attached an early draft of the ISO spec (from 2006) here:

fixed_point_C_spec

I don’t think the extensions simplify the use of fixed types very much. The programmer still needs to know how many bits are allocated to integer and fractional parts, and how the number and positions of bits may change (during multiplication for example). What the extensions do provide is a way to access the saturation and rounding modes of the processor without writing assembly code. With this level of access, it is possible to write much more efficient C code to handle these operations.

The advantages of C code over assembly are quicker coding and debugging, and more portable code (that is, code that can run on more than one type of processor). However, I noticed that details such as fixed point fractional points and handling of rounding are implementation dependent. So the portability may only be applicable for “similar” processors.

I have never coded anything using the stdfix.h definitions. As far as I can see, the GCC compiler and the Dinkumware libraries are the only tools using these extensions. I’m not sure if or when it will come into popular use, but it’s something to consider if one is coding fixed point math operations in C.

Rounding in Fixed Point Number Conversions

Posted August 19, 2009 by Shawn
Categories: dsp, fixed point

Tags: , , , ,

When converting from one fixed point representation to another, there is often a right shift operation to eliminate bits. (Or higher order bits are just stored without keeping the lower order bits.) This occurs when converting from a Q31 to a Q15 format number for example, since 16 bits need to be eliminated. Before throwing away the unused bits, sometimes it is desirable to perform a rounding operation first. This can improve the accuracy of results, and can prevent the introduction of a bias during conversion of a signal. Rounding is also an important operation when generating fixed point filter coefficients from floating point values, but that is not the subject of this post.

To illustrate rounding, I will use an example where six different signed Q7.8 numbers are converted to a signed Q15.0 number (a regular 16 bit integer). I will illustrate truncation (throwing away the least significant eight bits) and rounding. Recall that a Q7.8 number has seven integer bits and eight fractional bits. For the example, the six numbers will be 1.25, 1.5, 1.75, -1.25, -1.5 and -1.75.

The first thing to determine is how these numbers will be represented in a 16 bit integer register. Multiplying each by 256 (which is two to the power eight) gives the following result (in hexadecimal):

1.25 = 0x0140

1.5 = 0x0180

1.75 = 0x01C0

-1.25 = 0xFEC0

-1.5 = 0xFE80

-1.75 = 0xFE40

Now if the numbers are truncated, the result is found by shifting right by eight. Here are the results:

truncate(1.25) = 0x0001 = 1

truncate(1.5) = 0x0001 = 1

truncate(1.75) = 0x0001 = 1

truncate(-1.25) = 0xFFFE = -2

truncate(-1.5) = 0xFFFE = -2

truncate(-1.75) = 0xFFFE = -2

For the positive numbers, the result of truncation is that the fractional part is discarded. The negative number results are more interesting. The result is that the fractional part is lost, and the integer part has been reduced by one. If a series of these numbers had a mean of zero before truncation, then the series would have a mean of less than zero after truncation. Rounding is used to avoid this problem of introduced bias and to make results more accurate.

Truncation is not really the correct term for the example above. More accurately, a “floor” operation is being executed. A floor operation returns the greatest integer that is not greater than the operand.

In a common method of rounding, a binary one is added to the most significant bit of the bits that are to be thrown away. And then a truncation is performed. In the current example, we would add 0.5, represented as 128 decimal or 0x0080 in our 16 bit integer word. So the results in our example are as follows:

round(1.25) = (0x0140 + 0x80) >> 8 = 0x0001 = 1

round(1.5) = (0x0180 + 0x80) >> 8 = 0x0002 = 2

round(1.75) = (0x01C0 + 0x80) >> 8 = 0x0002 = 2

round(-1.25) = (0xFEC0 + 0x80) >> 8 = 0xFFFF = -1

round(-1.5) = (0xFE80+ 0x80) >> 8 = 0xFFFF = -1

round(-1.75) = (0xFE40 + 0x80) >> 8 = 0xFFFE = -2

These results are less problematic than using simple truncation, but there is still a bias due to the non-symmetry of the 1.5 and -1.5 cases. The amount of bias depends on the data set. Even if a set of data to be converted contained only positive values, there is still a bias introduced, because all of the values that end in exactly .5 are rounded to the next highest integer. One way to eliminate this bias is to round even and odd values differently (even and odd to the left of the rounding bit position).

For the more common conversion of Q31 to Q15 numbers, the rounding constant is one shifted left by fifteen, or 32768 decimal, or 0x8000 hexadecimal.

Some of the Texas Instrument DSPs have rounding instructions that can be performed on the accumulator register prior to saving a result to memory. For example, the TMS320C55x processor includes the ROUND instruction (full name is “round accumulator content”). The instruction has two different modes. The “biased” mode adds 0x8000 to the 40 bit accumulator register. The “unbiased” mode conditionally adds 0x8000 based on the value of the least significant 17 bits. It is designed to address the bias problems I described above. Wikipedia has a good discussion of rounding and bias errors (http://en.wikipedia.org/wiki/Rounding). The TMS320C55x is using the “round half to even” method of rounding for the unbiased mode, and “round half up” for the biased mode.

Although it seems simple on the surface, rounding in fixed point conversions has some important effects on the bias of resulting computations.

Overflow Handling in Fixed Point Computations

Posted August 10, 2009 by Shawn
Categories: fixed point

Tags: , , , ,

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.

Addition and Subtraction

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.

Blurry Equations

Posted August 4, 2009 by Shawn
Categories: Uncategorized

Tags: ,

I noticed that some of the equations look a bit blurry in my blog. If they look excessively blurry, try changing the web browser zoom value to default (ctrl+0 in Internet Explorer or Firefox). Also, the font for the C code examples may look small in some cases. If this is a problem, try clicking on the title of the post.

Fixed Point Division

Posted July 21, 2009 by Shawn
Categories: Uncategorized

I have created a new post to replace this one.  Click here to see it.

Fixed Point Multiplication

Posted July 14, 2009 by Shawn
Categories: fixed point

Tags: , , ,

Fixed point addition and subtraction are straightforward. Additions and subtractions are performed using integer operations. For example, if two 16 bit Q15 format numbers are added, the result is a Q15 number. But what about fixed about multiplication? What happens if two Q15 numbers are multiplied?

Let’s try an example. Take 0.5 multiplied by 0.25. In Q15 the number 0.5 is represented (in hexadecimal) as 0x8000 times 0.5 or 0x4000. Similarly, 0.25 is 0x2000. When we multiply these together, the product is 0x08000000. Obviously the result is not a Q15 number since the number of bits required is more than 16. The expected product, 0.125, is 0x1000 in Q15.

To see what is going on, define the following two Q15 numbers a and b:

a = \frac{\hat{a}}{{2}^{15}}

b = \frac{\hat{b}}{{2}^{15}}

where \hat{a} and \hat{b} are the integer representations of our numbers (0x4000 and 0x2000 in our example). The product of a and b is:

c = ab=\frac{\hat{a}\hat{b}}{{2}^{30}}

From the above, it can be seen that the product is a Q30 number. Going back to our example, 0x4000 times 0x2000 is 0x08000000, which is 0.125 times {2}^{30} .

A general rule when multiplying a Qm format number by a Qn format number, is that the product will be a Q(m+n) number. The number of bits required to represent the product is at least (n+m) for unsigned multiplication and (n+m+1) for signed (twos complement) multiplication.

For the more general case of a Qa.b number times a Qc.d number, the product is Q(a+c).(b+d).  The number of bits needed for the result is (a + b + c + d + 1) for signed numbers (and one less for unsigned numbers).

Consider the example of a Q16 unsigned multiplication between the two largest unsigned numbers that can be represented. The largest Q16 number is 65535/65536 = 0.9999847412109375. The product is 0xffff times 0xffff or 0xfffe0001. The result is a Q32 number requiring at least 32 bits. If we divide by {2}^{32} then we get 0.99996948265470564365386962890625, the expected result.

There are a number of things that are done with the product of a multiplication, depending on the application. Some of the commonly seen options are:

  1. Convert the product to a different Q format.

  2. Use the product in the resulting Q format.

  3. Add the product to a running sum in an accumulator register.

  4. Convert the product to a different Q format, then add to a running sum.

Let’s look at some of these options for the case of signed multiplication using Q15 format numbers. For case 1, assume we want to multiply two Q15 numbers and get a Q15 result. The required operation is to take the Q30 product, and shift it right by 15 bits. The result can then be stored in 16 bits. There is also the option of rounding the product before shifting out the lower 15 bits (I may discuss rounding in a future post). Some CPU architectures are better set up to shift the product left by 1, and then store the upper 16 bits. This is almost exactly the same as shifting right by 15 bits and keeping the lower 16 bits.

Multiply-accumulate (MAC) operations are used a lot in many DSP algorithms. Many processors have one or more dedicated accumulator registers for this purpose (often with 32 or 40 bits). For the case of Q15 multiplies, each Q30 product can be summed to the accumulator.

I have seen a lot of code that shifts each product left by 1 when performing the MAC operations. Some DSP chips can do the left shift in hardware using a special mode of the ALU. In this case, the value in the accumulator is in Q31 format. Although very common, this method has a greater chance of overflow problems since each product is effectively two times bigger. I think this method became popular because certain older DSP chip architectures required the storing of the high 16 bits of the accumulator or product register, rather than having a single cycle instruction allowing a shift by 15 bits.

In summary, because multiplication operations are often a chief component of signal processing implementations, it is important to understand how they work. This is especially true for fixed point operations, where one must know the effect of multiplication on the format of the numbers themselves.