## Design IIR Band-Reject Filters

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

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

IntroductionThis 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

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

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

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

IntroductionThis 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

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]=...## Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.

This article will consider the engineering of a self-calibration & self-test capability to enable the project hardware to be configured and its basic performance evaluated and verified, ready for the development of the low-latency controller DSP firmware and closed-loop applications. Performance specifications will be documented in due course, on the project website here.

- Part 6: Self-Calibration, Measurements and Signalling (this part)
- Part 5:

## Simplest Calculation of Half-band Filter Coefficients

Half-band filters are lowpass FIR filters with cut-off frequency of one-quarter of sampling frequency fs and odd symmetry about fs/4 [1]*. And it so happens that almost half of the coefficients are zero. The passband and stopband bandwiths are equal, making these filters useful for decimation-by-2 and interpolation-by-2. Since the zero coefficients make them computationally efficient, these filters are ubiquitous in DSP systems.

Here we will compute half-band...

## Learn About Transmission Lines Using a Discrete-Time Model

We don’t often think about signal transmission lines, but we use them every day. Familiar examples are coaxial cable, Ethernet cable, and Universal Serial Bus (USB). Like it or not, high-speed clock and signal traces on printed-circuit boards are also transmission lines.

While modeling transmission lines is in general a complex undertaking, it is surprisingly simple to model a lossless, uniform line with resistive terminations by using a discrete-time approach. A...

## Adaptive Beamforming is like Squeezing a Water Balloon

Adaptive beamforming was first developed in the 1960s for radar and sonar applications. The main idea is that signals can be captured using multiple sensors and the sensor outputs can be combined to enhance the signals propagating from specific directions and attenuate (null out) signals from other directions. It has grown immensely in recent years as processors have become faster and cheaper. Today, adaptive beamforming applications include smart speakers (like the Amazon Echo),...

## Wavelets II - Vanishing Moments and Spectral Factorization

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...

## Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved

This blog clarifies how to obtain and interpret the z-domain transfer function of the coupled-form 2nd-order IIR resonator. The coupled-form 2nd-order IIR resonator was developed to overcome a shortcoming in the standard 2nd-order IIR resonator. With that thought in mind, let's take a brief look at a standard 2nd-order IIR resonator.

Standard 2nd-Order IIR Resonator A block diagram of the standard 2nd-order IIR resonator is shown in Figure 1(a). You've probably seen that block diagram many...

## Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose spectrum contains replications of the original time sequence's spectrum.

Background

The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.

Figure 1

The '↑ L' operation in Figure 1 means to...

## Matlab Code to Synthesize Multiplierless FIR Filters

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...

## Correcting an Important Goertzel Filter Misconception

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:

"...is marginally stable and vulnerable tonumerical error accumulation when computed usinglow-precision arithmetic and...## Design IIR Band-Reject Filters

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...

## Simple Discrete-Time Modeling of Lossy LC Filters

There are many software applications that allow modeling LC filters in the frequency domain. But sometimes it is useful to have a time domain model, such as when you need to analyze a mixed analog and DSP system. For example, the system in Figure 1 includes an LC filter as well as a DSP portion. The LC filter could be an anti-alias filter, a channel filter, or some other LC network. For a design using undersampling, the filter would be bandpass [1]. By modeling...

## Exponential Smoothing with a Wrinkle

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in this blog article.

Basic Exponential Smoothing

Exponential smoothing is also known as...

## Is It True That *j* is Equal to the Square Root of -1 ?

A few days ago, on the YouTube.com web site, I watched an interesting video concerning complex numbers and the j operator. The video's author claimed that the statement "j is equal to the square root of negative one" is incorrect. What he said was:

He justified his claim by going through the following exercise, starting with:

Based on the algebraic identity:

the author rewrites Eq. (1) as:

If we assume

Eq. (3) can be rewritten...

## Differentiating and integrating discrete signals

I am back at work on Think DSP, adding a new chapter on differentiation and integration. In the previous chapter (which you can read here) I present Gaussian smoothing, show how smoothing in the time domain corresponds to a low-pass filter in the frequency domain, and present the Convolution Theorem.

In the current chapter, I start with the first difference operation (diff in Numpy) and show that it corresponds to a high-pass filter in the frequency domain. I use historical stock...

## Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved

This blog clarifies how to obtain and interpret the z-domain transfer function of the coupled-form 2nd-order IIR resonator. The coupled-form 2nd-order IIR resonator was developed to overcome a shortcoming in the standard 2nd-order IIR resonator. With that thought in mind, let's take a brief look at a standard 2nd-order IIR resonator.

Standard 2nd-Order IIR Resonator A block diagram of the standard 2nd-order IIR resonator is shown in Figure 1(a). You've probably seen that block diagram many...

## Multiplying Two Binary Numbers

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...

## 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...

## Sensors Expo - Trip Report & My Best Video Yet!

This was my first time at Sensors Expo and my second time in Silicon Valley and I must say I had a great time.

Before I share with you what I find to be, by far, my best 'highlights' video yet for a conference/trade show, let me try to entertain you with a few anecdotes from this trip. If you are not interested by my stories or maybe don't have the extra minutes needed to read them, please feel free to skip to the end of this blog post to watch the...

## Fitting a Damped Sine Wave

A damped sine wave is described by

$$ x_{(k)} = A \cdot e^{\alpha \cdot k} \cdot cos(\omega \cdot k + p)\tag{1}$$

with frequency $\omega$ , phase p , initial amplitude A and damping constant $\alpha$ . The $x_{(k)}$ are the samples of the function at equally spaced points in time.

With $x_{(k)}$ given, one often has to find the unknown parameters of the function. This can be achieved for instance with nonlinear approximation or with DFT – methods.

I present a method to find the...

## New Comments System (please help me test it)

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...

## 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...## A Narrow Bandpass Filter in Octave or Matlab

The design of a very narrow bandpass FIR filter, coded in either Octave or Matlab, can prove challenging if a computationally-efficient filter is required. This is especially true if the sampling rate is high relative to the filter's center frequency. The most obvious filter design methods, using either window-based or Remez ( Parks-McClellan ) functions, can easily result in filters with many thousands of taps.

The filter to be described reduces the computational effort (and thus...