Interpolator Design: Get the Stopbands Right
In this article, I present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands.
The Discrete Fourier Transform as a Frequency Response
The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1]. So, if we can find H(k) by whatever method, it should be identical to the DFT of h(n). In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n).
Consider the four-tap FIR filter in Figure 1, where each block labeled Ts represents a delay of one...
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
In this part, I’ll show how to design a Hilbert Transformer using the coefficients of a half-band filter as a starting point, which turns out to be remarkably simple. I’ll also show how a half-band filter can be synthesized using the Matlab function firpm, which employs the Parks-McClellan algorithm.
A half-band filter is a type of lowpass, even-symmetric FIR filter having an odd number of taps, with the even-numbered taps (except for the main tap) equal to zero. This...
Add the Hilbert Transformer to Your DSP Toolkit, Part 1
In some previous articles, I made use of the Hilbert transformer, but did not explain its theory in any detail. In this article, I’ll dig a little deeper into how the Hilbert Transformer works. Understanding the Hilbert Transformer involves a modest amount of mathematics, but the payoff in useful applications is worth it.
As we’ll learn, a Hilbert Transformer is just a particular type of Finite Impulse Response (FIR) filter. In Part 1 of this article, I’ll...
Candan's Tweaks of Jacobsen's Frequency Approximation
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by explaining how a tweak to a well known frequency approximation formula makes it better, and another tweak makes it exact. The first tweak is shown to be the first of a pattern and a novel approximation formula is made from the second. It only requires a few extra calculations beyond the original approximation to come up with an approximation suitable for most...
A Recipe for a Basic Trigonometry Table
IntroductionThis is an article that is give a better understanding to the Discrete Fourier Transform (DFT) by showing how to build a Sine and Cosine table from scratch. Along the way a recursive method is developed as a tone generator for a pure tone complex signal with an amplitude of one. Then a simpler multiplicative one. Each with drift correction factors. By setting the initial values to zero and one degrees and letting it run to build 45 values, the entire set of values needed...
Learn About Transmission Lines Using a Discrete-Time Model
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
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...
In Search of The Fourth Wave
Last year I participated in the first DSP Related online conference, where I presented a short talk called "In Search of The Fourth Wave". It's based on a small mystery I encountered when I was working on Think DSP. As you might know:
A sawtooth wave contains harmonics at integer multiples of the fundamental frequency, and their amplitudes drop off in proportion to 1/f. A square wave contains only odd multiples of the fundamental, but they also drop off...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.
Simplest Calculation of Half-band Filter Coefficients
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...
The Discrete Fourier Transform of Symmetric Sequences
Symmetric sequences arise often in digital signal processing. Examples include symmetric pulses, window functions, and the coefficients of most finite-impulse response (FIR) filters, not to mention the cosine function. Examining symmetric sequences can give us some insights into the Discrete Fourier Transform (DFT). An even-symmetric sequence is centered at n = 0 and xeven(n) = xeven(-n). The DFT of xeven(n) is real. Most often, signals we encounter start at n = 0, so they are not strictly speaking even-symmetric. We’ll look at the relationship between the DFT’s of such sequences and those of true even-symmetric sequences.
Design IIR Butterworth Filters Using 12 Lines of Code
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:
N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...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...
How the Cooley-Tukey FFT Algorithm Works | Part 1 - Repeating Calculations
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the Cooley-Tukey FFT algorithm works.
How the Cooley-Tukey FFT Algorithm Works | Part 4 - Twiddle Factors
The beauty of the FFT algorithm is that it does the same thing over and over again. It treats every stage of the calculation in exactly the same way. However, this. “one-size-fits-all” approach, although elegant and simple, causes a problem. It misaligns samples and introduces phase distortions during each stage of the algorithm. To overcome this, we need Twiddle Factors, little phase correction factors that push things back into their correct positions before continuing onto the next stage.
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...Learn to Use the Discrete Fourier Transform
Discrete-time sequences arise in many ways: a sequence could be a signal captured by an analog-to-digital converter; a series of measurements; a signal generated by a digital modulator; or simply the coefficients of a digital filter. We may wish to know the frequency spectrum of any of these sequences. The most-used tool to accomplish this is the Discrete Fourier Transform (DFT), which computes the discrete frequency spectrum of a discrete-time sequence. The DFT is easily calculated using software, but applying it successfully can be challenging. This article provides Matlab examples of some techniques you can use to obtain useful DFT’s.
Plotting Discrete-Time Signals
A discrete-time sinusoid can have frequency up to just shy of half the sample frequency. But if you try to plot the sinusoid, the result is not always recognizable. For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine. But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph, which does not look like a sine when you connect the dots. We typically want the plot of a...
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...
Simplest Calculation of Half-band Filter Coefficients
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...
Interpolation Basics
This article covers interpolation basics, and provides a numerical example of interpolation of a time signal. Figure 1 illustrates what we mean by interpolation. The top plot shows a continuous time signal, and the middle plot shows a sampled version with sample time Ts. The goal of interpolation is to increase the sample rate such that the new (interpolated) sample values are close to the values of the continuous signal at the sample times [1]. For example, if...
Plotting Discrete-Time Signals
A discrete-time sinusoid can have frequency up to just shy of half the sample frequency. But if you try to plot the sinusoid, the result is not always recognizable. For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine. But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph, which does not look like a sine when you connect the dots. We typically want the plot of a...
An s-Plane to z-Plane Mapping Example
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.
...Phase or Frequency Shifter Using a Hilbert Transformer
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.
Phase ShifterA conceptual diagram of a phase shifter is shown in Figure 1, where the bold lines indicate complex...
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...
Evaluate Window Functions for the Discrete Fourier Transform
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...
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...
The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase
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