Digital PLL's -- Part 1

Neil Robertson June 7, 201616 comments
1. Introduction

Figure 1.1 is a block diagram of a digital PLL (DPLL).  The purpose of the DPLL is to lock the phase of a numerically controlled oscillator (NCO) to a reference signal.  The loop includes a phase detector to compute phase error and a loop filter to set loop dynamic performance.  The output of the loop filter controls the frequency and phase of the NCO, driving the phase error to zero.

One application of the DPLL is to recover the timing in a digital...


Decimator Image Response

Neil Robertson May 24, 20164 comments

Note:  this is an improved version of a post I made to the dsp forum a few weeks ago.

This article presents a way to compute and plot the image response of a decimator.  I’m defining the image response as the unwanted spectrum of the impulse response after downsampling, relative to the desired passband response. 

Consider a decimate-by-4 filter with fs= 4 Hz, to which we apply the signal spectrum shown in Figure 1.  The desired signal is the CW component at 0.22 Hz,...


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...


Filter a Rectangular Pulse with no Ringing

Neil Robertson May 12, 201610 comments

To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients:  they must all be positive.  However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical.  What we are describing is basically a window function.

Consider a rectangular pulse 32 samples long with fs = 1 kHz.  Here is the Matlab code to generate the pulse:

N= 64; fs= 1000; % Hz sample...

ADC Clock Jitter Model, Part 2 – Random Jitter

Neil Robertson April 22, 20184 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,...


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...


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...


Use Matlab Function pwelch to Find Power Spectral Density – or Do It Yourself

Neil Robertson January 13, 2019

In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1].  These include windowing the signal, taking the magnitude-squared of the DFT, and computing the vector of frequencies.  The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the so-called Welch power spectral density estimate [3,4].

In this article, I’ll present some...