## Fractional Delay FIR Filters

Consider the following Finite Impulse Response (FIR) coefficients:

b = [b0 b1 b2 b1 b0]

These coefficients form a 5-tap symmetrical FIR filter having constant group delay [1,2] over 0 to fs/2 of:

D = (ntaps – 1)/2 = 2      samples

For a symmetrical filter with an odd number of taps, the group delay is always an integer number of samples, while for one with an even number of taps, the group delay is always an integer + 0.5 samples.  Can we design a filter...

## The DFT of Finite-Length Time-Reversed Sequences

Recently I've been reading papers on underwater acoustic communications systems and this caused me to investigate the frequency-domain effects of time-reversal of time-domain sequences. I created this blog because there is so little coverage of this topic in the literature of DSP.

This blog reviews the two types of time-reversal of finite-length sequences and summarizes their discrete Fourier transform (DFT) frequency-domain characteristics.

The Two Types of Time-Reversal in DSP

...

## Model Signal Impairments at Complex Baseband

In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise.  To provide concrete examples, we’ll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code.  Although our example system is QAM, the models can be used for any complex-baseband signal.

I used a very simple complex-baseband model of a QAM system in my last

## Update To: A Wide-Notch Comb Filter

December 9, 2019

This blog presents alternatives to the wide-notch comb filter described in Reference [1]. That comb filter, which for notational reasons I now call a 2-RRS wide notch comb filter, is shown in Figure 1. I use the "2-RRS" moniker because the comb filter uses two recursive running sum (RRS) networks.

The z-domain transfer function of the 2-RRS wide-notch comb filter, H2-RRS(z), is:

References

[1] R. Lyons, "A Wide-Notch Comb Filter", dsprelated.com Blogs, Nov. 24, 2019, Available...

## A Wide-Notch Comb Filter

This blog describes a linear-phase comb filter having wider stopband notches than a traditional comb filter.

Background

Let's first review the behavior of a traditional comb filter. Figure 1(a) shows a traditional comb filter comprising two cascaded recursive running sum (RRS) comb filters. Figure 1(b) shows the filter's co-located dual poles and dual zeros on the z-plane, while Figure 1(c) shows the filter's positive-frequency magnitude response when, for example, D = 9. The...

## An Efficient Lowpass Filter in Octave

November 6, 2019

This article describes an efficient linear-phase lowpass FIR filter, coded using the Octave programming language. The intention is to focus on the implementation in software, but references are provided for those who wish to undertake further study of interpolated FIR filters [1]- [3].

The input signal is processed as a vector of samples (eg from a .wav file), which are converted to a matrix format.   The complete filter is thus referred to as a Matrix IFIR or...

## Compute Modulation Error Ratio (MER) for QAM

This post defines the Modulation Error Ratio (MER) for QAM signals, and shows how to compute it.  As we’ll see, in the absence of impairments other than noise, the MER tracks the signal’s Carrier-to-Noise Ratio (over a limited range).  A Matlab script at the end of the PDF version of this post computes MER for a simplified QAM-64 system.

Figure 1 is a simplified block diagram of a QAM system.  The transmitter includes a source of QAM symbols, a root-Nyquist...

## Polynomial calculations on an FIR filter engine, part 1

October 1, 20191 comment

Polynomial evaluation is structurally akin to FIR filtering and fits dedicated filtering engines quite well, with certain caveats. It’s a technique that has wide applicability. This two-part note discusses transducer and amplifier non-linearity compensation, function approximation and aspects of harmonic signal synthesis.

Need for polynomials as general non-linear functions

Many transducer types exhibit a non-linear relationship between a measured parameter, such as a voltage, and...

## The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance

This blog shows the danger in evaluating the performance of a digital integration network based solely on its frequency response curve. If you plan on implementing a digital integrator in your signal processing work I recommend you continue reading this blog.

Background

Typically when DSP practitioners want to predict the accuracy performance of a digital integrator they compare how closely that integrator's frequency response matches the frequency response of an ideal integrator [1,2]....

## Plotting Discrete-Time Signals

A discrete-time sinusoid can have frequency up to just shy of half the sample frequency.  But if you try to plot the sinusoid, the result is not always recognizable.  For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine.  But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph, which does not look like a sine when you connect the dots.  We typically want the plot of a...

## Design IIR Butterworth Filters Using 12 Lines of Code

While there are plenty of canned functions to design Butterworth IIR filters [1], it’s instructive and not that complicated to design them from scratch.  You can do it in 12 lines of Matlab code.  In this article, we’ll create a Matlab function butter_synth.m to design lowpass Butterworth filters of any order.  Here is an example function call for a 5th order filter:

N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...

## Computing FFT Twiddle Factors

Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2 fast Fourier transform (FFT).

Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to compute the individual twiddle factors of an N-point decimation-in-frequency...

## Python scipy.signal IIR Filtering: An Example

May 19, 2013
Introduction

In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ).  In this post I am going to conclude the IIR filter design review with an example.

Previous posts:

## Free DSP Books on the Internet

While surfing the "net" I have occasionally encountered signal processing books whose chapters could be downloaded to my computer. I started keeping a list of those books and, over the years, that list has grown to over forty books. Perhaps the list will be of interest to you.

## Minimum Shift Keying (MSK) - A Tutorial

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...

## Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm

July 7, 20151 comment

If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.

Preliminaries To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are described by:

$$Forward \ FFT \rightarrow X(m) = \sum_{n=0}^{N-1} x(n)e^{-j2\pi nm/N} \tag{1}$$  Inverse \ FFT \rightarrow x(n) = {1 \over N} \sum_{m=0}^{N-1}...

## The DFT Magnitude of a Real-valued Cosine Sequence

This blog may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.

To be specific, if we perform an N-point DFT on N real-valued time-domain samples of a discrete cosine wave, having exactly integer k cycles over N time samples, the peak magnitude of the cosine wave's...

## Python scipy.signal IIR Filter Design

Introduction

The following is an introduction on how to design an infinite impulse response (IIR) filters using the Python scipy.signal package.  This post, mainly, covers how to use the scipy.signal package and is not a thorough introduction to IIR filter design.  For complete coverage of IIR filter design and structure see one of the references.

Filter Specification

Before providing some examples lets review the specifications for a filter design.  A filter...

## Delay estimation by FFT

Given x=sig(t) and y=ref(t), returns [c, ref(t+delta), delta)] = fitSignal(y, x);:Estimates and corrects delay and scaling factor between two signals Code snippet

This article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.

Introduction

There are many DSP-related problems, where an unknown timing between two signals needs to be determined and corrected, for example, radar, sonar,...

## Understanding and Relating Eb/No, SNR, and other Power Efficiency Metrics

Introduction

Evaluating the performance of communication systems, and wireless systems in particular, usually involves quantifying some performance metric as a function of Signal-to-Noise-Ratio (SNR) or some similar measurement. Many systems require performance evaluation in multipath channels, some in Doppler conditions and other impairments related to mobility. Some have interference metrics to measure against, but nearly all include noise power as an impairment. Not all systems are...

## Crowdfunding Articles?

Many of you have the knowledge and talent to write technical articles that would benefit the EE community.  What is missing for most of you though, and very understandably so, is the time and motivation to do it.

But what if you could make some money to compensate for your time spent on writing the article(s)?  Would some of you find the motivation and make the time?

I am thinking of implementing a system/mechanism that would allow the EE community to...

## Embedded World 2018 - More Videos!

March 27, 20181 comment

After the interview videos last week, this week I am very happy to release two more videos taken at Embedded World 2018 and that I am proud of.

For both videos, I made extensive use of my two new toys, a Zhiyun Crane Gimbal and a Sony a6300 camera.

The use of a gimbal like the Zhiyun makes a big difference in terms of making the footage look much more stable and cinematographic.

As for the Sony camera, it takes fantastic slow-motion footage and...

## Embedded World 2018 - The Interviews

March 21, 2018

Once again this year, I had the chance to go to Embedded World in Nuremberg Germany.  And once again this year, I brought my video equipment to try and capture some of the most interesting things at the show.

Something new this year, I asked Jacob Beningo if he would partner with me in doing interviews with a few vendors.  I would operate the camera while Jacob would ask the right questions to the vendors to make them talk about the key products/features that...

## Finally got a drone!

As a reader of my blog, you already know that I have been making videos lately and thoroughly enjoying the process.  When I was in Germany early this summer (and went 280 km/h in a porsche!) to produce SEGGER's 25th anniversary video, the company bought a drone so we could get an aerial shot of the party (at about the 1:35 mark in this video).  Since then, I have been obsessing on buying a drone for myself and finally made the move a few weeks ago - I acquired a used DJI...

## SEGGER's 25th Anniversary Video

Chances are you will find this video more interesting to watch if you take five minutes to first read the story of the week I spent at SEGGER's headquarters at the end of June.

The video is only a little more than 2 minutes long.  If you decide to watch it, make sure to go full screen and I would really love to read your thoughts about it in the comments down bellow.  Do you think a video like this succeeds in making the viewer want to learn more about the company?...

## Went 280km/h (174mph) in a Porsche Panamera in Germany!

Those of you who've been following my blog lately already know that I am going through some sort of mid-life crisis that involves going out there to meet people and make videos.  It all started with Embedded World early this year, then continued at ESC Boston a couple of months ago and the latest chapter just concluded as I returned from Germany after spending a week at SEGGER's headquarters to produce a video to highlight their 25th anniversary.

## Going back to Germany!

A couple of blog posts ago, I wrote that the decision to go to ESC Boston ended up being a great one for many different reasons.  I came back from the conference energized and really happy that I went.

These feelings were amplified a few days after my return when I received an email from Rolf Segger, the founder of SEGGER Microcontroller (check out their very new website), asking if I would be interested in visiting their headquarters...

## ESC Boston's Videos are Now Up

June 5, 2017

In my last blog, I told you about my experience at ESC Boston and the few videos that I was planning to produce and publish.  Here they are, please have a look and any feedback (positive or negative) is appreciated.

Short Highlight

This is a very short (one minute) montage of some of the footage that I shot at the show & conference.  In future shows, I absolutely need to insert clips here and there of engineers saying a few words about the conference (why they...

## Back from ESC Boston

NOT going to ESC Boston would have allowed me to stay home, in my comfort zone.

NOT going to ESC Boston would have saved me from driving in the absolutely horrible & stressful Boston traffic1.

NOT going to ESC Boston would have saved me from having to go through a full search & questioning session at the Canada Customs on my return2.

2017/06/06 update: Videos are now up!

So two days...