## Frequency Translation by Way of Lowpass FIR Filtering

February 4, 20175 comments

Some weeks ago a question appeared on the dsp.related Forum regarding the notion of translating a signal down in frequency and lowpass filtering in a single operation [1]. It is possible to implement such a process by embedding a discrete cosine sequence's values within the coefficients of a traditional lowpass FIR filter. I first learned about this process from Reference [2]. Here's the story.

Traditional Frequency Translation Prior To Filtering

Think about the process shown in...

## Minimum Shift Keying (MSK) - A Tutorial

January 25, 20174 comments

Minimum Shift Keying (MSK) is one of the most spectrally efficient modulation schemes available. Due to its constant envelope, it is resilient to non-linear distortion and was therefore chosen as the modulation technique for the GSM cell phone standard.

MSK is a special case of Continuous-Phase Frequency Shift Keying (CPFSK) which is a special case of a general class of modulation schemes known as Continuous-Phase Modulation (CPM). It is worth noting that CPM (and hence CPFSK) is a...

## New Video: Parametric Oscillations

January 4, 2017

I just posted this last night.  It's kinda off-topic from the mission of the channel, but I realized that it had been months since I'd posted a video, and having an excuse to build on helped keep me on track.

## Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

November 22, 20163 comments

Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.

One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...

## Some Thoughts on Sampling

November 15, 20162 comments

Some time ago, I came across an interesting problem. In the explanation of sampling process, a representation of impulse sampling shown in Figure 1 below is illustrated in almost every textbook on DSP and communications. The question is: how is it possible that during sampling, the frequency axis gets scaled by $1/T_s$ -- a very large number? For an ADC operating at 10 MHz for example, the amplitude of the desired spectrum and spectral replicas is $10^7$! I thought that there must be...

## Matlab Code to Synthesize Multiplierless FIR Filters

October 31, 20162 comments

This article presents Matlab code to synthesize multiplierless Finite Impulse Response (FIR) lowpass filters.

A filter coefficient can be represented as a sum of powers of 2.  For example, if a coefficient = decimal 5 multiplies input x, the output is $y= 2^2*x + 2^0*x$.  The factor of $2^2$ is then implemented with a shift of 2 bits.  This method is not efficient for coefficients having a lot of 1’s, e.g. decimal 31 = 11111.  To reduce the number of non-zero...

## Wavelets II - Vanishing Moments and Spectral Factorization

October 11, 2016

In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.

Besides the content of the last post, you should be familiar with basic complex algebra, the...

## Fibonacci trick

October 10, 20164 comments

I'm working on a video, tying the Fibonacci sequence into the general subject of difference equations.

Here's a fun trick: take any two consecutive numbers in the Fibonacci sequence, say 34 and 55.  Now negate one and use them as the seed for the Fibonacci sequence, larger magnitude first, i.e.

$-55, 34, \cdots$

Carry it out, and you'll eventually get the Fibonacci sequence, or it's negative:

$-55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1 \cdots$

This is NOT a general property of difference...

## The Power Spectrum

October 8, 2016

Often, when calculating the spectrum of a sampled signal, we are interested in relative powers, and we don’t care about the absolute accuracy of the y axis.  However, when the sampled signal represents an analog signal, we sometimes need an accurate picture of the analog signal’s power in the frequency domain.  This post shows how to calculate an accurate power spectrum.

Parseval’s theorem [1,2] is a property of the Discrete Fourier Transform (DFT) that...

October 4, 201618 comments

I thought it would take me a day or two to implement, it took almost two weeks...

But here it is, the new comments systems for blogs, heavily inspired by the forum system I developed earlier this year.

Which means that:

• You can easily add images, either by drag and drop or through the 'Insert Image' button
• You can add MathML, TeX and ASCIImath equations and they will be rendered with Mathjax
• You can add code snippets and they will be highlighted with highlights.js
• You can edit...

## Waveforms that are their own Fourier Transform

January 16, 200812 comments

Mea Culpa

There are many scary things about writing a technical book. Can I make the concepts clear? It is worth the effort? Will it sell? But all of these pale compared to the biggest fear: What if I'm just plain wrong? Not being able to help someone is one thing, but leading them astray is far worse.

My book on DSP has now been published for almost ten years. I've found lots of typos, a few misstatements, and many places where the explanations confuse even me. But I have been lucky;...

## 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 =...

## A Simple Complex Down-conversion Scheme

January 21, 20085 comments
Recently I was experimenting with complex down-conversion schemes. That is, generating an analytic (complex) version, centered at zero Hz, of a real bandpass signal that was originally centered at ±fs/4 (one fourth the sample rate). I managed to obtain one such scheme that is computationally efficient, and it might be of some mild interest to you guys. The simple complex down-conversion scheme is shown in Figure 1(a).

It works like this: say we have a real xR(n) input bandpass...

## The Swiss Army Knife of Digital Networks

June 13, 20163 comments

This blog describes a general discrete-signal network that appears, in various forms, inside so many DSP applications.

Figure 1 shows how the network's structure has the distinct look of a digital filter—a comb filter followed by a 2nd-order recursive network. However, I do not call this useful network a filter because its capabilities extend far beyond simple filtering. Through a series of examples I've illustrated the fundamental strength of this Swiss Army Knife of digital networks...

## The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase

August 18, 201516 comments

This blog discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this blog answers the question:

What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?

I'll declare two things to convince you to continue reading.

Declaration# 1: "That the coefficients must be symmetrical" is not a correct

## An s-Plane to z-Plane Mapping Example

September 24, 20166 comments

While surfing around the Internet recently I encountered the 's-plane to z-plane mapping' diagram shown in Figure 1. At first I thought the diagram was neat because it's a good example of the old English idiom: "A picture is worth a thousand words." However, as I continued to look at Figure 1 I began to detect what I believe are errors in the diagram.

Reader, please take a few moments to see if you detect any errors in Figure 1.

...

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

October 22, 20126 comments

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...

## 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...

## How to Find a Fast Floating-Point atan2 Approximation

May 26, 20177 comments
Context Over a short period of time, I came across nearly identical approximations of the two parameter arctangent function, atan2, developed by different companies, in different countries, and even in different decades. Fascinated with how the coefficients used in these approximations were derived, I set out to find them. This atan2 implementation is based around a rational approximation of arctangent on the domain -1 to 1:

 atan(z) \approx \dfrac{z}{1.0 +...

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

August 18, 2012
Overview
• Octaveforge / Matlab design script. Download: here
• 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...