## The Power Spectrum

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...

## Digital PLL's -- Part 2

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...## Digital PLL's -- Part 1

1. IntroductionFigure 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...

## Peak to Average Power Ratio and CCDF

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

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...## Add a Power Marker to a Power Spectral Density (PSD) Plot

Perhaps we should call most Power Spectral Density (PSD) calculations relative PSD, because usually we don’t have to worry about absolute power levels. However, for cases (e.g., measurements or simulations) where we are concerned with absolute power, it would be nice to be able to display it on a PSD plot. Unfortunately, you can’t read the power directly from the plot. For example, the plotted spectral peak of a narrowband signal, such as a sinewave, is lower than the...

## There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle

Reference [1] has some 3D plots of frequency response magnitude above the unit circle in the Z-plane. I liked them enough that I wrote a Matlab function to plot the response of any digital filter this way. I’m not sure how useful these plots are, but they’re fun to look at. The Matlab code is listed in the Appendix.This post is available in PDF format for easy...

## Coefficients of Cascaded Discrete-Time Systems

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...

## Digital Filter Instructions from IKEA?

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Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and play.

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and play.

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and...

## Filter a Rectangular Pulse with no Ringing

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...## 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...

## Multiplierless Half-band Filters and Hilbert Transformers

This article provides coefficients of multiplierless Finite Impulse Response 7-tap, 11-tap, and 15-tap half-band filters and Hilbert Transformers. Since Hilbert transformer coefficients are simply related to half-band coefficients, multiplierless Hilbert transformers are easily derived from multiplierless half-bands.

## DAC Zero-Order Hold Models

This article provides two simple time-domain models of a DAC’s zero-order hold. These models will allow us to find time and frequency domain approximations of DAC outputs, and simulate analog filtering of those outputs. Developing the models is also a good way to learn about the DAC ZOH function.

## Model a Sigma-Delta DAC Plus RC Filter

Sigma-delta digital-to-analog converters (SD DAC’s) are often used for discrete-time signals with sample rate much higher than their bandwidth. For the simplest case, the DAC output is a single bit, so the only interface hardware required is a standard digital output buffer. Because of the high sample rate relative to signal bandwidth, a very simple DAC reconstruction filter suffices, often just a one-pole RC lowpass. In this article, I present a simple Matlab function that models the combination of a basic SD DAC and one-pole RC filter. This model allows easy evaluation of the overall performance for a given input signal and choice of sample rate, R, and C.

## Decimators Using Cascaded Multiplierless Half-band Filters

In my last post, I provided coefficients for several multiplierless half-band FIR filters. In the comment section, Rick Lyons mentioned that such filters would be useful in a multi-stage decimator. For such an application, any subsequent multipliers save on resources, since they operate at a fraction of the maximum sample frequency. We’ll examine the frequency response and aliasing of a multiplierless decimate-by-8 cascade in this article, and we’ll also discuss an interpolator cascade using the same half-band filters.