Design IIR Filters Using Cascaded Biquads

Neil Robertson February 11, 201824 comments

This article shows how to implement a Butterworth IIR lowpass filter as a cascade of second-order IIR filters, or biquads.  We’ll derive how to calculate the coefficients of the biquads and do some examples using a Matlab function biquad_synth provided in the Appendix.  Although we’ll be designing Butterworth filters, the approach applies to any all-pole lowpass filter (Chebyshev, Bessel, etc).  As we’ll see, the cascaded-biquad design is less sensitive to coefficient...

Design IIR Highpass Filters

Neil Robertson February 3, 20182 comments

This post is the fourth in a series of tutorials on IIR Butterworth filter design.  So far we covered lowpass [1], bandpass [2], and band-reject [3] filters; now we’ll design highpass filters.  The general approach, as before, has six steps:

Find the poles of a lowpass analog prototype filter with Ωc = 1 rad/s. Given the -3 dB frequency of the digital highpass filter, find the corresponding frequency of the analog highpass filter (pre-warping). Transform the...

Design IIR Band-Reject Filters

Neil Robertson January 17, 20182 comments

In this post, I show how to design IIR Butterworth band-reject filters, and provide two Matlab functions for band-reject filter synthesis.  Earlier posts covered IIR Butterworth lowpass [1] and bandpass [2] filters.  Here, the function br_synth1.m designs band-reject filters based on null frequency and upper -3 dB frequency, while br_synth2.m designs them based on lower and upper -3 dB frequencies.   I’ll discuss the differences between the two approaches later in this...

Design IIR Bandpass Filters

Neil Robertson January 6, 201810 comments

In this post, I present a method to design Butterworth IIR bandpass filters.  My previous post [1] covered lowpass IIR filter design, and provided a Matlab function to design them.  Here, we’ll do the same thing for IIR bandpass filters, with a Matlab function bp_synth.m.  Here is an example function call for a bandpass filter based on a 3rd order lowpass prototype:

N= 3; % order of prototype LPF fcenter= 22.5; % Hz center frequency, Hz bw= 5; ...

Phase and Amplitude Calculation for a Pure Complex Tone in a DFT

Cedron Dawg January 6, 2018

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from a DFT bin value and knowing the frequency. This is a much simpler problem to solve than the corresponding case for a pure real tone which I covered in an earlier blog article[1]. In the noiseless single tone case, these equations will be exact. In the presence of noise or other tones...

Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators

Steve Maslen January 5, 20187 comments
This article will look at some DSP Sine-wave oscillators and will show how an FPGA with limited floating-point performance due to latency, can be persuaded to produce much higher sample-rate sine-waves of high quality. 

Comparisons will be made between implementations on Intel Cyclone V and Cyclone 10 GX FPGAs. An Intel numerically controlled oscillator

Linear Feedback Shift Registers for the Uninitiated, Part XII: Spread-Spectrum Fundamentals

Jason Sachs December 29, 20171 comment

Last time we looked at the use of LFSRs for pseudorandom number generation, or PRNG, and saw two things:

  • the use of LFSR state for PRNG has undesirable serial correlation and frequency-domain properties
  • the use of single bits of LFSR output has good frequency-domain properties, and its autocorrelation values are so close to zero that they are actually better than a statistically random bit stream

The unusually-good correlation properties...

An Efficient Linear Interpolation Scheme

Rick Lyons December 27, 201725 comments

This blog presents a computationally-efficient linear interpolation trick that requires at most one multiply per output sample.

Background: Linear Interpolation

Looking at Figure 1(a) let's assume we have two points, [x(0),y(0)] and [x(1),y(1)], and we want to compute the value y, on the line joining those two points, associated with the value x. 

       Figure 1: Linear interpolation: given x, x(0), x(1), y(0), and y(1), compute the value of y. ...

An Alternative Form of the Pure Real Tone DFT Bin Value Formula

Cedron Dawg December 17, 2017

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving alternative exact formulas for the bin values of a real tone in a DFT. The derivation of the source equations can be found in my earlier blog article titled "DFT Bin Value Formulas for Pure Real Tones"[1]. The new form is slighty more complicated and calculation intensive, but it is more computationally accurate in the vicinity of near integer frequencies. This...

Design IIR Butterworth Filters Using 12 Lines of Code

Neil Robertson December 10, 201712 comments

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]=...

The Swiss Army Knife of Digital Networks

Rick Lyons June 13, 20168 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...

Discrete Wavelet Transform Filter Bank Implementation (part 1)

David October 27, 20101 comment

UPDATE: Added graphs and code to explain the frequency division of the branches

The focus of this article is to briefly explain an implementation of this transform and several filter bank forms. Theoretical information about DWT can be found elsewhere.

First of all, a 'quick and dirty' simplified explanation of the differences between DFT and DWT:

The DWT (Discrete Wavelet Transform), simply put, is an operation that receives a signal as an input (a vector of data) and...

Online DSP Classes: Why Such a High Dropout Rate?

Rick Lyons October 7, 201718 comments

Last year the IEEE Signal Processing Magazine published a lengthy article describing three university-sponsored online digital signal processing (DSP) courses [1]. The article detailed all the effort the professors expended in creating those courses and the courses' perceived values to students. 

However, one fact that struck me as important, but not thoroughly addressed in the article, was the shocking dropout rate of those online courses. For two of the courses the article's...

Dealing With Fixed Point Fractions

Mike January 5, 20163 comments

Fixed point fractional representation always gives me a headache because I screw it up the first time I try to implement an algorithm. The difference between integer operations and fractional operations is in the overflow.  If the representation fits in the fixed point result, you can not tell the difference between fixed point integer and fixed point fractions.  When integers overflow, they lose data off the most significant bits.  When fractions overflow, they lose data off...

Should DSP Undergraduate Students Study z-Transform Regions of Convergence?

Rick Lyons September 14, 201613 comments

Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:

     "I noticed that you didn't discuss z-plane regions of      convergence here. In my undergraduate DSP class we      spent a lot of classroom and homework time on the  ...

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

Stephane Boucher July 10, 201712 comments

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.  

Correcting an Important Goertzel Filter Misconception

Rick Lyons July 6, 201512 comments

Recently I was on the Signal Processing Stack Exchange web site (a question and answer site for DSP people) and I read a posted question regarding Goertzel filters [1]. One of the subscribers posted a reply to the question by pointing interested readers to a Wikipedia web page discussing Goertzel filters [2]. I noticed the Wiki web site stated that a Goertzel filter:

" marginally stable and vulnerable tonumerical error accumulation when computed usinglow-precision arithmetic and...

Amplitude modulation and the sampling theorem

Allen Downey December 18, 20156 comments

I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College.  He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.

Most of the code for the chapter is done: you can check it out in this IPython notebook.  I haven't written the text yet, but I'll outline it here, and paste in the key figures.


Some Observations on Comparing Efficiency in Communication Systems

Eric Jacobsen March 17, 2011

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...

Multiplying Two Binary Numbers

Rick Lyons March 16, 20117 comments

I just encountered what I think is an interesting technique for multiplying two integer numbers. Perhaps some of the readers here will also find it interesting.

Here's the technique: assume we want to multiply 18 times 17. We start by writing 18 and 17, side-by-side in column A and column B, as shown at the top of Figure 1. Next we divide the 18 at the top of column A by two, retaining only the integer part of the division, and double the 17 at the top of column B. The results of those two...