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...


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...


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...

Design IIR Band-Reject Filters

Neil Robertson January 17, 2018

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; ...

Design IIR Butterworth Filters Using 12 Lines of Code

Neil Robertson December 10, 201711 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:


The Power Spectrum

Neil Robertson 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...


Matlab Code to Synthesize Multiplierless FIR Filters

Neil Robertson October 31, 20163 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...


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...


IIR Bandpass Filters Using Cascaded Biquads

Neil Robertson April 20, 20196 comments

In an earlier post [1], we implemented lowpass IIR filters using a cascade of second-order IIR filters, or biquads.  This post provides a Matlab function to do the same for Butterworth bandpass IIR filters.  Compared to conventional implementations, bandpass filters based on biquads are less sensitive to coefficient quantization [2].  This becomes important when designing narrowband filters.

A biquad section block diagram using the Direct Form II structure [3,4] is shown in...


Evaluate Window Functions for the Discrete Fourier Transform

Neil Robertson December 18, 2018

The Discrete Fourier Transform (DFT) operates on a finite length time sequence to compute its spectrum.  For a continuous signal like a sinewave, you need to capture a segment of the signal in order to perform the DFT.  Usually, you also need to apply a window function to the captured signal before taking the DFT [1 - 3].  There are many different window functions and each produces a different approximation of the spectrum.  In this post, we’ll present Matlab code that...


Design IIR Band-Reject Filters

Neil Robertson January 17, 2018

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...


Canonic Signed Digit (CSD) Representation of Integers

Neil Robertson February 18, 2017

In my last post I presented Matlab code to synthesize multiplierless FIR filters using Canonic Signed Digit (CSD) coefficients.  I included a function dec2csd1.m (repeated here in Appendix A) to convert decimal integers to binary CSD values.  Here I want to use that function to illustrate a few properties of CSD numbers.

In a binary signed-digit number system, we allow each binary digit to have one of the three values {0, 1, -1}.  Thus, for example, the binary value 1 1...


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...


Modeling a Continuous-Time System with Matlab

Neil Robertson June 6, 20172 comments

Many of us are familiar with modeling a continuous-time system in the frequency domain using its transfer function H(s) or H(jω).  However, finding the time response can be challenging, and traditionally involves finding the inverse Laplace transform of H(s).  An alternative way to get both time and frequency responses is to transform H(s) to a discrete-time system H(z) using the impulse-invariant transform [1,2].  This method provides an exact match to the continuous-time...


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,...