## Take Control of Noise with Spectral Averaging

Most engineers have seen the moment-to-moment fluctuations that are common with instantaneous measurements of a supposedly steady spectrum. You can see these fluctuations in magnitude and phase for each frequency bin of your spectrogram. Although major variations are certainly reason for concern, recall that we don’t live in an ideal, noise-free world. After verifying the integrity of your measurement setup by checking connections, sensors, wiring, and the like, you might conclude that the...

## How precise is my measurement?

Some might argue that measurement is a blend of skepticism and faith. While time constraints might make you lean toward faith, some healthy engineering skepticism should bring you back to statistics. This article reviews some practical statistics that can help you satisfy one common question posed by skeptical engineers: “How precise is my measurement?” As we’ll see, by understanding how to answer it, you gain a degree of control over your measurement time.

An accurate, precise...## Feedback Controllers - Making Hardware with Firmware. Part 8. Control Loop Test-bed

This part in the series will consider the signals, measurements, analyses and configurations for testing high-speed low-latency feedback loops and their controllers. Along with basic test signals, a versatile IFFT signal generation scheme will be discussed and implemented. A simple controller under test will be constructed to demonstrate the analysis principles in preparation for the design and evaluation of specific controllers and closed-loop applications.

Additional design...## Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from several DFT bin values and knowing the frequency. This article is functionally an extension of my prior article "Phase and Amplitude Calculation for a Pure Complex Tone in a DFT"[1] which used only one bin for a complex tone, but it is actually much more similar to my approach for real...

## Phase and Amplitude Calculation for a Pure Complex Tone in a DFT

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from a DFT bin value and knowing the frequency. This is a much simpler problem to solve than the corresponding case for a pure real tone which I covered in an earlier blog article[1]. In the noiseless single tone case, these equations will be exact. In the presence of noise or other tones...

## Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators

This article will look at some DSP Sine-wave oscillators and will show how an FPGA with limited floating-point performance due to latency, can be persuaded to produce much higher sample-rate sine-waves of high quality.Comparisons will be made between implementations on Intel Cyclone V and Cyclone 10 GX FPGAs. An Intel numerically controlled oscillator

## An Efficient Linear Interpolation Scheme

This blog presents a computationally-efficient linear interpolation trick that requires at most one multiply per output sample.

Background: Linear Interpolation

Looking at Figure 1(a) let's assume we have two points, [x(0),y(0)] and [x(1),y(1)], and we want to compute the value y, on the line joining those two points, associated with the value x.

Figure 1: Linear interpolation: given x, x(0), x(1), y(0), and y(1), compute the value of y. ...

## An Alternative Form of the Pure Real Tone DFT Bin Value Formula

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving alternative exact formulas for the bin values of a real tone in a DFT. The derivation of the source equations can be found in my earlier blog article titled "DFT Bin Value Formulas for Pure Real Tones"[1]. The new form is slighty more complicated and calculation intensive, but it is more computationally accurate in the vicinity of near integer frequencies. This...

## Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.

This article will consider the engineering of a self-calibration & self-test capability to enable the project hardware to be configured and its basic performance evaluated and verified, ready for the development of the low-latency controller DSP firmware and closed-loop applications. Performance specifications will be documented in due course, on the project website here.

- Part 6: Self-Calibration, Measurements and Signalling (this part)
- Part 5:

## Improved Three Bin Exact Frequency Formula for a Pure Real Tone in a DFT

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by extending the exact two bin formulas for the frequency of a real tone in a DFT to the three bin case. This article is a direct extension of my prior article "Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT"[1]. The formulas derived in the previous article are also presented in this article in the computational order, rather than the indirect order they were...

## The DFT Output and Its Dimensions

The Discrete Fourier Transform, or DFT, converts a signal from discrete time to discrete frequency. It is commonly implemented as and used as the Fast Fourier Transform (FFT). This article will attempt to clarify the format of the DFT output and how it is produced.

Living in the real world, we deal with real signals. The data we typically sample does not have an imaginary component. For example, the voltage sampled by a receiver is a real value at a particular point in time. Let’s...

## A Fast Real-Time Trapezoidal Rule Integrator

This blog presents a computationally-efficient network for computing real‑time discrete integration using the Trapezoidal Rule.

Background

While studying what is called "N-sample Romberg integration" I noticed that such an integration process requires the computation of many individual smaller‑sized integrations using the Trapezoidal Rule integration method [1]. My goal was to create a computationally‑fast real‑time Trapezoidal Rule integration network to increase the processing...

## Should DSP Undergraduate Students Study z-Transform Regions of Convergence?

Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:

"I noticed that you didn't discuss z-plane regions of convergence here. In my undergraduate DSP class we spent a lot of classroom and homework time on the ...

## Signed serial-/parallel multiplication

Keywords: Binary signed multiplication implementation, RTL, Verilog, algorithm

Summary- A detailed discussion of bit-level trickstery in signed-signed multiplication
- Algorithm based on Wikipedia example
- Includes a Verilog implementation with parametrized bit width

A straightforward method to multiply two binary numbers is to repeatedly shift the first argument a, and add to a register if the corresponding bit in the other argument b is set. The...

## Exact Frequency Formula for a Pure Real Tone in a DFT

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a real tone in a DFT. According to current teaching, this is not possible, so this article should be considered a major theoretical advance in the discipline. The formula is presented in a few different formats. Some sample calculations are provided to give a numerical demonstration of the formula in use. This article is...

## Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...

## Pentagon Construction Using Complex Numbers

A method for constructing a pentagon using a straight edge and a compass is deduced from the complex values of the Fifth Roots of Unity. Analytic values for the points are also derived.

## Frequency Dependence in Free Space Propagation

Introduction

It seems to be fairly common knowledge, even among practicing professionals, that the efficiency of propagation of wireless signals is frequency dependent. Generally it is believed that lower frequencies are desirable since pathloss effects will be less than they would be at higher frequencies. As evidence of this, the Friis Transmission Equation[i] is often cited, the general form of which is usually written as:

Pr = Pt Gt Gr ( λ / 4πd )2 (1)

where the...

## Discrete Wavelet Transform Filter Bank Implementation (part 1)

UPDATE: Added graphs and code to explain the frequency division of the branches

The focus of this article is to briefly explain an implementation of this transform and several filter bank forms. Theoretical information about DWT can be found elsewhere.

First of all, a 'quick and dirty' simplified explanation of the differences between DFT and DWT:

The DWT (Discrete Wavelet Transform), simply put, is an operation that receives a signal as an input (a vector of data) and...

## Discrete-Time PLLs, Part 1: Basics

In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.

## The DFT Output and Its Dimensions

The Discrete Fourier Transform, or DFT, converts a signal from discrete time to discrete frequency. It is commonly implemented as and used as the Fast Fourier Transform (FFT). This article will attempt to clarify the format of the DFT output and how it is produced.

Living in the real world, we deal with real signals. The data we typically sample does not have an imaginary component. For example, the voltage sampled by a receiver is a real value at a particular point in time. Let’s...

## Python scipy.signal IIR Filter Design Cont.

In the previous post the Python scipy.signal iirdesign function was disected. We reviewed the basics of filter specification and reviewed how to use the iirdesign function to design IIR filters. The previous post I only demonstrated low pass filter designs. The following are examples how to use the iirdesign function for highpass, bandpass, and stopband filters designs.

Highpass FilterThe following is a highpass filter design for the different filter...

## Understanding Radio Frequency Distortion

OverviewThe topic of this article are the effects of radio frequency distortions on a baseband signal, and how to model them at baseband. Typical applications are use as a simulation model or in digital predistortion algorithms.

IntroductionTransmitting and receiving wireless signals usually involves analog radio frequency circuits, such as power amplifiers in a transmitter or low-noise amplifiers in a receiver.Signal distortion in those circuits deteriorates the link quality. When...

## Design of an anti-aliasing filter for a DAC

Overview- Octaveforge / Matlab design script. Download: here
- weighted numerical optimization of Laplace-domain transfer function
- linear-phase design, optimizes vector error (magnitude and phase)
- design process calculates and corrects group delay internally
- includes sinc() response of the sample-and-hold stage in the ADC
- optionally includes multiplierless FIR filter

Digital-to-analog conversion connects digital...

## Instantaneous Frequency Measurement

I would like to talk about the oft used method of measuring the carrier frequency in the world of Signal Collection and Characterization world. It is an elegant technique because of its simplicity. But, of course, with simplicity, there come drawbacks (sometimes...especially with this one!).

In the world of Radar detection and characterization, one of the key characteristics of interest is the carrier frequency of the signal. If the radar is pulsed, you will have a very wide bandwidth, a...

## Discrete Wavelet Transform Filter Bank Implementation (part 1)

UPDATE: Added graphs and code to explain the frequency division of the branches

The focus of this article is to briefly explain an implementation of this transform and several filter bank forms. Theoretical information about DWT can be found elsewhere.

First of all, a 'quick and dirty' simplified explanation of the differences between DFT and DWT:

The DWT (Discrete Wavelet Transform), simply put, is an operation that receives a signal as an input (a vector of data) and...

## Should DSP Undergraduate Students Study z-Transform Regions of Convergence?

Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:

"I noticed that you didn't discuss z-plane regions of convergence here. In my undergraduate DSP class we spent a lot of classroom and homework time on the ...

## Dealing With Fixed Point Fractions

Fixed point fractional representation always gives me a headache because I screw it up the first time I try to implement an algorithm. The difference between integer operations and fractional operations is in the overflow. If the representation fits in the fixed point result, you can not tell the difference between fixed point integer and fixed point fractions. When integers overflow, they lose data off the most significant bits. When fractions overflow, they lose data off...

## Amplitude modulation and the sampling theorem

I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.

Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key figures.

Convolution...

## Sampling bandpass signals

Sampling bandpass signals 1.1 IntroductionIt is known [1], [3] that bandpass signals can be sampled with a sampling frequency which is lower than the sampling frequency according to the sampling theorem.

Fig. 1 shows an example of how the spectrum of a bandpass signal sampled with $f_s$ (Fig. 1a) arises in the baseband with $−f_s / 2 ≤ f < f_s/2$. The bandpass signal is assumed to have a center frequency $f_c = (f_{max} + f_{min})/2$ and bandwidth $\Delta f...