Which one?

1. 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;

2. Introduction to Digital Filtering SMD077 – Computer Architecture 31-Oct-2001 Dennis M. Akos Luleå University of Technology

3. 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

4. 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

5. Finite Impulse Response (FIR) Filter Input Sampled Signal x[n] Output (Filtered) Sampled Signal y[n] FIR Filter • 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?

6. Example: 3rd Order FIR Filter Structure Input Sampled Signal x[n] Perfect structure for SIMD (Single-Instruction Multiple-Data) operations x[n - 1] x[n - 2] x[n - 3] delay delay delay • 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 • bn’s are constants and completely define how the filter will act on the input (lowpass, highpass, …) x x x x b0 b1 b2 b3 Output (Filtered) Sampled Signal y[n] + + +

7. x[0] x[2] x[4] x[5] x[3] x[1] FIR Filter Input & Output Sequences Sampled Input Signal Resulting Output Signal y0] y[2] … … y[4] y[5] • 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 y[3] y[1] transient portion time time

8. FIR Filter Resulting Algorithm from “C Algorithms for Real-Time DSP” by P. Embree • /************************************************************************** • 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) • 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); • }