## Decimator Image Response

Note:  this is an improved version of a post I made to the dsp forum a few weeks ago.

This article presents a way to compute and plot the image response of a decimator.  I’m defining the image response as the unwanted spectrum of the impulse response after downsampling, relative to the desired passband response.

Consider a decimate-by-4 filter with fs= 4 Hz, to which we apply the signal spectrum shown in Figure 1.  The desired signal is the CW component at 0.22 Hz,...

## Peak to Average Power Ratio and CCDF

Peak to Average Power Ratio (PAPR) is often used to characterize digitally modulated signals.  One example application is setting the level of the signal in a digital modulator.  Knowing PAPR allows setting the average power to a level that is just low enough to minimize clipping.

However, for a random signal, PAPR is a statistical quantity.  We have to ask, what is the probability of a given peak power?  Then we can decide where to set the average...

## Filter a Rectangular Pulse with no Ringing

To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients:  they must all be positive.  However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical.  What we are describing is basically a window function.

Consider a rectangular pulse 32 samples long with fs = 1 kHz.  Here is the Matlab code to generate the pulse:

N= 64; fs= 1000; % Hz sample...

## Data Types for Control & DSP

There's a lot of information out there on what data types to use for digital signal processing, but there's also a lot of confusion, so the topic bears repeating.

I recently posted an entry on PID control. In that article I glossed over the data types used by showing "double" in all of my example code.  Numerically, this should work for most control problems, but it can be an extravagant use of processor resources.  There ought to be a better way to determine what precision you need...

## PID Without a PhD

I both consult and teach in the area of digital control. Through both of these efforts, I have found that while there certainly are control problems that require all the expertise I can bring to bear, there are a great number of control problems that can be solved with the most basic knowledge of simple controllers, without resort to any formal control theory at all.

This article will tell you how to implement a simple controller in software and how to tune it without getting into heavy...

## Digital Envelope Detection: The Good, the Bad, and the Ugly

Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.

Here I focus on envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope)...

## Harmonic Notch Filter

My basement is covered with power lines and florescent lights which makes collecting ECG and EEG data  rather difficult due to the 60 cycle hum.  I found the following notch filter to work very well at eliminating the background signal without effecting the highly amplified signals I was looking for.

The notch filter is based on the a transfer function with the form $$H(z)=\frac{1}{2}(1+A(z))$$ where A(z) is an all pass filter. The original paper [1] describes a method to...

## A Useful Source of Signal Processing Information

I just discovered a useful web-based source of signal processing information that was new to me. I thought I'd share what I learned with the subscribers here on DSPRelated.com.

The Home page of the web site that I found doesn't look at all like it would be useful to us DSP fanatics. But if you enter some signal processing topic of interest, say, "FM demodulation" (without the quotation marks) into the 'Search' box at the top of the web page

and click the red 'SEARCH...

## 3 Good News

March 9, 20161 comment
Good News #1

Last week, I announced a new and ambitious reward program that will be funded by the new Vendors Directory.

This week, I am happy to announce that we have our firsts two sponsors!  Quantum Leaps & Abelon Systems have agreed to pay the sponsorship fee to be listed in the new Vendors Directory.  Because of their support, there is now some money in the reward pool (\$1,000) and enough to pay for the firsts 500 'beers' awarded.  Please...

## Padé Delay is Okay Today

This article is going to be somewhat different in that I’m not really writing it for the typical embedded systems engineer. Rather it’s kind of a specialized topic, so don’t be surprised if you get bored and move on to something else. That’s fine by me.

Anyway, let’s just jump ahead to the punchline. Here’s a numerical simulation of a step response to a $p=126, q=130$ Padé approximation of a time delay:

Impressed? Maybe you should be. This...

Introduction

Many DSP problems have close ties with the analog world. For example, a switched-mode audio power amplifier uses a digital control loop to open and close power transistors driving an analog filter. There are commercial tools for digital-analog cosimulation: Simulink comes to mind, and mainstream EDA vendors support VHDL-AMS or Verilog-A in their...

## Goertzel Algorithm for a Non-integer Frequency Index

October 7, 2013

If you've read about the Goertzel algorithm, you know it's typically presented as an efficient way to compute an individual kth bin result of an N-point discrete Fourier transform (DFT). The integer-valued frequency index k is in the range of zero to N-1 and the standard block diagram for the Goertzel algorithm is shown in Figure 1. For example, if you want to efficiently compute just the 17th DFT bin result (output sample X17) of a 64-point DFT you set integer frequency index k = 17 and N =...

## Some Observations on Comparing Efficiency in Communication Systems

March 17, 2011
Introduction

Engineering is usually about managing efficiencies of one sort or another. One of my favorite working definitions of an engineer says, "An engineer is somebody who can do for a nickel what any damn fool can do for a dollar." In that case, the implication is that the cost is one of the characteristics being optimized. But cost isn't always the main efficiency metric, or at least the only one. Consider how a common transportation appliance, the automobile, is optimized...

## Design of an anti-aliasing filter for a DAC

August 18, 2012
Overview
• weighted numerical optimization of Laplace-domain transfer function
• linear-phase design, optimizes vector error (magnitude and phase)
• design process calculates and corrects group delay internally
• includes sinc() response of the sample-and-hold stage in the ADC
• optionally includes multiplierless FIR filter
Problem Figure 1: Typical FIR-DAC-analog lowpass line-up

Digital-to-analog conversion connects digital...

## Signed serial-/parallel multiplication

February 16, 2014

Keywords: Binary signed multiplication implementation, RTL, Verilog, algorithm

Summary
• A detailed discussion of bit-level trickstery in signed-signed multiplication
• Algorithm based on Wikipedia example
• Includes a Verilog implementation with parametrized bit width
Signed serial-/parallel multiplication

A straightforward method to multiply two binary numbers is to repeatedly shift the first argument a, and add to a register if the corresponding bit in the other argument b is set. The...

## The Exponential Nature of the Complex Unit Circle

Introduction

This is an article to hopefully give an understanding to Euler's magnificent equation:

$$e^{i\theta} = cos( \theta ) + i \cdot sin( \theta )$$

This equation is usually proved using the Taylor series expansion for the given functions, but this approach fails to give an understanding to the equation and the ramification for the behavior of complex numbers. Instead an intuitive approach is taken that culminates in a graphical understanding of the equation.

Complex...

## Setting the 3-dB Cutoff Frequency of an Exponential Averager

This blog discusses two ways to determine an exponential averager's weighting factor so that the averager has a given 3-dB cutoff frequency. Here we assume the reader is familiar with exponential averaging lowpass filters, also called a "leaky integrators", to reduce noise fluctuations that contaminate constant-amplitude signal measurements. Exponential averagers are useful because they allow us to implement lowpass filtering at a low computational workload per output sample.

Figure 1 shows...

## Spline interpolation

A cookbook recipe for segmented y=f(x) 3rd-order polynomial interpolation based on arbitrary input data. Includes Octave/Matlab design script and Verilog implementation example. Keywords: Spline, interpolation, function modeling, fixed point approximation, data fitting, Matlab, RTL, Verilog

Introduction

Splines describe a smooth function with a small number of parameters. They are well-known for example from vector drawing programs, or to define a "natural" movement path through given...

## DSP Related Math: Nice Animated GIFs

I was browsing the ECE subreddit lately and found that some of the most popular posts over the last few months have been animated GIFs helping understand some mathematical concepts.  I thought there would be some value in aggregating the DSP related gifs on one page.

The relationship between sin, cos, and right triangles: Constructing a square wave with infinite series (see this...