Tutorials

Design a DAC sinx/x Corrector

Neil Robertson July 22, 20187 comments

This post provides a Matlab function that designs linear-phase FIR sinx/x correctors.  It includes a table of fixed-point sinx/x corrector coefficients for different DAC frequency ranges.

A sinx/x corrector is a digital (or analog) filter used to compensate for the sinx/x roll-off inherent in the digital to analog conversion process.  In DSP math, we treat the digital signal applied to the DAC is a sequence of impulses.  These are converted by the DAC into contiguous pulses...


Digital PLL’s, Part 3 – Phase Lock an NCO to an External Clock

Neil Robertson May 27, 201823 comments

Sometimes you may need to phase-lock a numerically controlled oscillator (NCO) to an external clock that is not related to the system clocks of your ASIC or FPGA.  This situation is shown in Figure 1.  Assuming your system has an analog-to-digital converter (ADC) available, you can sync to the external clock using the scheme shown in Figure 2.  This time-domain PLL model is similar to the one presented in Part 1 of this series on digital PLL’s [1].  In that PLL, we...


ADC Clock Jitter Model, Part 2 – Random Jitter

Neil Robertson April 22, 20185 comments

In Part 1, I presented a Matlab function to model an ADC with jitter on the sample clock, and applied it to examples with deterministic jitter.  Now we’ll investigate an ADC with random clock jitter, by using a filtered or unfiltered Gaussian sequence as the jitter source.  What we are calling jitter can also be called time jitter, phase jitter, or phase noise.  It’s all the same phenomenon.  Typically, we call it jitter when we have a time-domain representation,...


ADC Clock Jitter Model, Part 1 – Deterministic Jitter

Neil Robertson April 16, 20182 comments

Analog to digital converters (ADC’s) have several imperfections that affect communications signals, including thermal noise, differential nonlinearity, and sample clock jitter [1, 2].  As shown in Figure 1, the ADC has a sample/hold function that is clocked by a sample clock.  Jitter on the sample clock causes the sampling instants to vary from the ideal sample time.  This transfers the jitter from the sample clock to the input signal.

In this article, I present a Matlab...


How precise is my measurement?

Sam Shearman March 28, 20183 comments

Some might argue that measurement is a blend of skepticism and faith. While time constraints might make you lean toward faith, some healthy engineering skepticism should bring you back to statistics. This article reviews some practical statistics that can help you satisfy one common question posed by skeptical engineers: “How precise is my measurement?” As we’ll see, by understanding how to answer it, you gain a degree of control over your measurement time.

An accurate, precise...

Phase or Frequency Shifter Using a Hilbert Transformer

Neil Robertson March 25, 201819 comments

In this article, we’ll describe how to use a Hilbert transformer to make a phase shifter or frequency shifter.  In either case, the input is a real signal and the output is a real signal.  We’ll use some simple Matlab code to simulate these systems.  After that, we’ll go into a little more detail on Hilbert transformer theory and design. 

This article is available in PDF format for easy printing.

Phase Shifter

A conceptual diagram...


Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins

Cedron Dawg March 14, 2018
Introduction

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 several DFT bin values and knowing the frequency. This article is functionally an extension of my prior article "Phase and Amplitude Calculation for a Pure Complex Tone in a DFT"[1] which used only one bin for a complex tone, but it is actually much more similar to my approach for real...


Coefficients of Cascaded Discrete-Time Systems

Neil Robertson March 4, 2018

In this article, we’ll show how to compute the coefficients that result when you cascade discrete-time systems.  With the coefficients in hand, it’s then easy to compute the time or frequency response.  The computation presented here can also be used to find coefficients of mixed discrete-time and continuous-time systems, by using a discrete time model of the continuous-time portion [1].

This article is available in PDF format for...


Design IIR Filters Using Cascaded Biquads

Neil Robertson February 11, 201817 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...

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

Rick Lyons 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

Rick Lyons 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.

...

Design IIR Filters Using Cascaded Biquads

Neil Robertson February 11, 201817 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...

Simplest Calculation of Half-band Filter Coefficients

Neil Robertson November 20, 20179 comments

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...


Discrete-Time PLLs, Part 1: Basics

Reza Ameli December 1, 20159 comments

Design Files: Part1.slx

Hi everyone,

In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.

In the first part of the series, we will be reviewing the basics of continuous-time baseband PLLs and we will see some useful mathematics that will give us insight into the inners working of PLLs. In the second part, we will focus on...


Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1

Cedron Dawg May 21, 2015
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...


Peak to Average Power Ratio and CCDF

Neil Robertson May 17, 20162 comments

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...


Wavelets II - Vanishing Moments and Spectral Factorization

Vincent Herrmann 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...


Digital PLL's -- Part 2

Neil Robertson June 15, 2016

In Part 1, we found the time response of a 2nd order PLL with a proportional + integral (lead-lag) loop filter.  Now let’s look at this PLL in the Z-domain [1, 2].  We will find that the response is characterized by a loop natural frequency ωn and damping coefficient ζ. 

Having a Z-domain model of the DPLL will allow us to do three things:

Compute the values of loop filter proportional gain KL and integrator gain KI that give the desired loop natural frequency and...

Wavelets I - From Filter Banks to the Dilation Equation

Vincent Herrmann September 28, 20169 comments

This is the first in what I hope will be a series of posts about wavelets, particularly about the Fast Wavelet Transform (FWT). The FWT is extremely useful in practice and also very interesting from a theoretical point of view. Of course there are already plenty of resources, but I found them tending to be either simple implementation guides that do not touch on the many interesting and sometimes crucial connections. Or they are highly mathematical and definition-heavy, for a...