## 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...

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

November 11, 2022
Introduction

This 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

October 4, 2022
Introduction

This 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

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...

## 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...

## 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...

## Third-Order Distortion of a Digitally-Modulated Signal

June 9, 2020
Analog designers are always harping about amplifier third-order distortion.  Why?  In this article, we’ll look at why third-order distortion is important, and simulate a QAM signal with third-order distortion.

In the following analysis, we assume that signal phase at the amplifier output is not a function of amplitude.  With this assumption, the output y of a non-ideal amplifier can be written as a power series of the input signal x: