## Evaluate Noise Performance of Discrete-Time Differentiators

When it comes to noise, all differentiators are not created equal.  Figure 1 shows the magnitude response of two differentiators.  They both have a useful bandwidth of a little less than π/8 radians (based on maximum magnitude response error of 2%).  Suppose we apply a signal with Gaussian noise to each of these differentiators.  The sinusoidal signal with noise is shown in the top of Figure 2.  Signal frequency is π/12.5 radians.  The output of the so-called...

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

January 12, 2022

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...

## The Discrete Fourier Transform and the Need for Window Functions

The Discrete Fourier Transform (DFT) is used to find the frequency spectrum of a discrete-time signal.  A computationally efficient version called the Fast Fourier Transform (FFT) is normally used to calculate the DFT.  But, as many have found to their dismay, the FFT, when used alone, usually does not provide an accurate spectrum.  The reason is a phenomenon called spectral leakage.

Spectral leakage can be reduced drastically by using a window function in conjunction...

## Modeling Anti-Alias Filters

September 26, 2021

Digitizing a signal using an Analog to Digital Converter (ADC) usually requires an anti-alias filter, as shown in Figure 1a.  In this post, we’ll develop models of lowpass Butterworth and Chebyshev anti-alias filters, and compute the time domain and frequency domain output of the ADC for an example input signal.  We’ll also model aliasing of Gaussian noise.  I hope the examples make the textbook explanations of aliasing seem a little more real.  Of course, modeling of...

## Digital Filter Instructions from IKEA?

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 play.

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

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

## Setting Carrier to Noise Ratio in Simulations

April 11, 2021

When simulating digital receivers, we often want to check performance with added Gaussian noise.  In this article, I’ll derive the simple equations for the rms noise level needed to produce a desired carrier to noise ratio (CNR or C/N).  I also provide a short Matlab function to generate a noise vector of the desired level for a given signal vector.

Definition of C/N

The Carrier to noise ratio is defined as the ratio of average signal power to noise power for a modulated...

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

February 7, 2021

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...

## Find Aliased ADC or DAC Harmonics (with animation)

When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics.  If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias.  In this case, it is not at once obvious if a given spur is a harmonic, and if so, its order.  In this article, we’ll present Matlab code to simulate the data converter nonlinearities and find the harmonic alias frequencies.  Note that Analog Devices has an online tool for...

## Compute Images/Aliases of CIC Interpolators/Decimators

Cascade-Integrator-Comb (CIC) filters are efficient fixed-point interpolators or decimators.  For these filters, all coefficients are equal to 1, and there are no multipliers.  They are typically used when a large change in sample rate is needed.  This article provides two very simple Matlab functions that can be used to compute the spectral images of CIC interpolators and the aliases of CIC decimators.

1.  CIC Interpolators

Figure 1 shows three interpolate-by-M...

## Design Square-Root Nyquist Filters

July 13, 2020

In his book on multirate signal processing, harris presents a nifty technique for designing square-root Nyquist FIR filters with good stopband attenuation [1].  In this post, I describe the method and provide a Matlab function for designing the filters.  You can find a Matlab function by harris for designing the filters at [2].

Background

Single-carrier modulation, such as QAM, uses filters to limit the bandwidth of the signal.  Figure 1 shows a simplified QAM system block...

## Second Order Discrete-Time System Demonstration

Discrete-time systems are remarkable:  the time response can be computed from mere difference equations, and the coefficients ai, bi of these equations are also the coefficients of H(z).  Here, I try to illustrate this remarkableness by converting a continuous-time second-order system to an approximately equivalent discrete-time system.  With a discrete-time model, we can then easily compute the time response to any input.  But note that the goal here is as much to...

## The Power Spectrum

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...

## Design IIR Highpass Filters

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...

## 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...

## A Simplified Matlab Function for Power Spectral Density

In an earlier post [1], I showed how to compute power spectral density (PSD) of a discrete-time signal using the Matlab function pwelch [2].  Pwelch is a useful function because it gives the correct output, and it has the option to average multiple Discrete Fourier Transforms (DFTs).  However, a typical function call has five arguments, and it can be hard to remember how to set them all and how they default.

In this post, I create a simplified PSD function by putting a...

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...

## 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...

## Fractional Delay FIR Filters

Consider the following Finite Impulse Response (FIR) coefficients:

b = [b0 b1 b2 b1 b0]

These coefficients form a 5-tap symmetrical FIR filter having constant group delay [1,2] over 0 to fs/2 of:

D = (ntaps – 1)/2 = 2      samples

For a symmetrical filter with an odd number of taps, the group delay is always an integer number of samples, while for one with an even number of taps, the group delay is always an integer + 0.5 samples.  Can we design a filter...