Which one?

• You have a vector, a[ ], of 50000 random integers, which can modern CPUs do faster and why?

//find max of vector of random ints max=0; for (inda=0; inda<50000;inda++) { if (a[inda] > max) { max = a[inda]; index=inda; } } //find avg of vector of random ints sum=0; for (inda=0; inda<50000;inda++) { sum = sum + a[inda]; } avg = ((double) sum) / 50000;

Introduction to Digital Filtering

SMD077 – Computer Architecture 31-Oct-2001 Dennis M. Akos

Luleå University of Technology

•

Motivation

Digital filtering is the “application” or “algorithm” that will be used in the majority of the labs • • • • Very representative programmable processor operation that has wide ranging real world applications This is a computer architecture course (not a course in programming or signal processing!?!) – Goal is to map algorithms to the hardware • Requires comprehensive understanding of the hardware, or architecture, itself • Compiler support does not exist, or is limited, for specialized hardware – Few will be designing programmable processor (definitely an option) but many will be using programmable processors Labs will be based around a Finite Impulse Response (FIR) Filter Basic understanding is achieved via time/frequency domain transforms

What this lecture is and is not!

• This is

not

a comprehensive overview of digital filters

– “Gloss over” much of the mathematics and theory involved with design and implementation of filters – Many good references are available

• It is a simple introduction to motivate/help you to better understand the upcoming labs

• • •

Finite Impulse Response (FIR) Filter

Input Sampled Signal x[n] FIR Filter y[n]  k M   0 b k  x[n  k] Output (Filtered) Sampled Signal y[n] Digital filter operates on a stream, or vector, of data representing some continuous signal – Sampled sinusoid – Audio signal (compact disk) There are four basic filter implementations: lowpass, highpass, bandpass, and bandstop (as well as many different classes (FIR, IIR, …) and subclasses (Butterworth, Chevychev,…) It is easiest to examine and consider the impact of different types of filters by their frequency domain characteristics – Consider the “audio equalizer” analogy – What is the frequency domain representation of the sinusoid? Sum of sinusoids?

Input Sampled Signal

x[n]

Example: 3 rd Order FIR Filter Structure delay

x[n - 1]

delay

x[n - 2]

delay

x[n - 3]

Perfect structure for SIMD (Single-Instruction Multiple-Data) operations

b 0

x

b 1

x

b 2

x

b 3

x + + +

Output (Filtered) Sampled Signal

y[n]

• • • FIR Filters can be of arbitrary order and extendable to an indefinite number of elements Filter order trade-off – Higher order results in sharper transitions between pass and stop bands – Higher order is more computationally complex b n ’s are constants and completely define how the filter will act on the input (lowpass, highpass, …)

FIR Filter Input & Output Sequences

Sampled Input Signal Resulting Output Signal

x[0] x[1] x[2] x[3] x[4] x[5] … y0] y[1] y[2] y[3] y[4] y[5] …

time transient portion time

• • Input signal can be specified as a vector of the resulting samples Note that there can be a “transient” in the output until the filter has all delay slots filled – Has implications for filtering short sequences – Higher order filters will have a longer transient

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • FIR Filter Resulting Algorithm

/************************************************************************** fir_filter - Perform fir filtering sample by sample on floats Requires array of filter coefficients and pointer to history.

Returns one output sample for each input sample.

float fir_filter(float input,float *coef,int n,float *history) from “C Algorithms for Real Time DSP” by P. Embree float input new float input sample float *coef pointer to filter coefficients int n number of coefficients in filter float *history history array pointer Returns float value giving the current output.

*************************************************************************/ float fir_filter(float input,float *coef,int n,float *history) { int i; float *hist_ptr,*hist1_ptr,*coef_ptr; float output; hist_ptr = history; hist1_ptr = hist_ptr; /* use for history update */ coef_ptr = coef + n - 1; /* point to last coef */ /* form output accumulation */ output = *hist_ptr++ * (*coef_ptr--); for(i = 2 ; i < n ; i++) { *hist1_ptr++ = *hist_ptr; /* update history array */ output += (*hist_ptr++) * (*coef_ptr--); } output += input * (*coef_ptr); /* input tap */ *hist1_ptr = input; /* last history */ return(output); }