## 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.

This article is available in PDF format for easy printing.

Phase ShifterA conceptual diagram...

## 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...

## 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...

## Design IIR Filters Using Cascaded Biquads

This article shows how to implement a Butterworth IIR lowpass filter as a cascade of second-order IIR filters, or biquads. We’ll derive how to calculate the coefficients of the biquads and do some examples using a Matlab function biquad_synth provided in the Appendix. Although we’ll be designing Butterworth filters, the approach applies to any all-pole lowpass filter (Chebyshev, Bessel, etc). As we’ll see, the cascaded-biquad design is less sensitive to coefficient...## 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...## Design IIR Band-Reject Filters

In this post, I show how to design IIR Butterworth band-reject filters, and provide two Matlab functions for band-reject filter synthesis. Earlier posts covered IIR Butterworth lowpass [1] and bandpass [2] filters. Here, the function br_synth1.m designs band-reject filters based on null frequency and upper -3 dB frequency, while br_synth2.m designs them based on lower and upper -3 dB frequencies. I’ll discuss the differences between the two approaches later in this...

## Design IIR Bandpass Filters

In this post, I present a method to design Butterworth IIR bandpass filters. My previous post [1] covered lowpass IIR filter design, and provided a Matlab function to design them. Here, we’ll do the same thing for IIR bandpass filters, with a Matlab function bp_synth.m. Here is an example function call for a bandpass filter based on a 3rd order lowpass prototype:

N= 3; % order of prototype LPF fcenter= 22.5; % Hz center frequency, Hz bw= 5; ...## 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...

## 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...

## 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:

## Compute the Frequency Response of a Multistage Decimator

Figure 1a shows the block diagram of a decimation-by-8 filter, consisting of a low-pass finite impulse response (FIR) filter followed by downsampling by 8 [1]. A more efficient version is shown in Figure 1b, which uses three cascaded decimate-by-two filters. This implementation has the advantages that only FIR 1 is sampled at the highest sample rate, and the total number of filter taps is lower.

The frequency response of the single-stage decimator before downsampling is just...

## 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...

## 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.

This article is available in PDF format for easy printing.

Phase ShifterA conceptual diagram...

## 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...

## Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT

One of the basic DSP principles states that a sampled time signal has a periodic spectrum with period equal to the sample rate. The derivation of can be found in textbooks [1,2]. You can also demonstrate this principle numerically using the Discrete Fourier Transform (DFT).

The DFT of the sampled signal x(n) is defined as:

$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \qquad (1)$$

Where

X(k) = discrete frequency spectrum of time sequence x(n)

## 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...## 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 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...

## Wavelets II - Vanishing Moments and Spectral Factorization

In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.

Besides the content of the last post, you should be familiar with basic complex algebra, the...

## 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

## 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.

...## Design IIR Bandpass Filters

In this post, I present a method to design Butterworth IIR bandpass filters. My previous post [1] covered lowpass IIR filter design, and provided a Matlab function to design them. Here, we’ll do the same thing for IIR bandpass filters, with a Matlab function bp_synth.m. Here is an example function call for a bandpass filter based on a 3rd order lowpass prototype:

N= 3; % order of prototype LPF fcenter= 22.5; % Hz center frequency, Hz bw= 5; ...## Discrete-Time PLLs, Part 1: Basics

Design Files: Part1.slx

Hi everyone,

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.

In the first part of the series, we will be reviewing the basics of continuous-time baseband PLLs and we will see some useful mathematics that will give us insight into the inners working of PLLs. In the second part, we will focus on...

## 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...

## Design IIR Filters Using Cascaded Biquads

This article shows how to implement a Butterworth IIR lowpass filter as a cascade of second-order IIR filters, or biquads. We’ll derive how to calculate the coefficients of the biquads and do some examples using a Matlab function biquad_synth provided in the Appendix. Although we’ll be designing Butterworth filters, the approach applies to any all-pole lowpass filter (Chebyshev, Bessel, etc). As we’ll see, the cascaded-biquad design is less sensitive to coefficient...## 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...

## 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...

## 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 frequency and...## Wavelets II - Vanishing Moments and Spectral Factorization

In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.

Besides the content of the last post, you should be familiar with basic complex algebra, the...

## Wavelets I - From Filter Banks to the Dilation Equation

This is the first in what I hope will be a series of posts about wavelets, particularly about the Fast Wavelet Transform (FWT). The FWT is extremely useful in practice and also very interesting from a theoretical point of view. Of course there are already plenty of resources, but I found them tending to be either simple implementation guides that do not touch on the many interesting and sometimes crucial connections. Or they are highly mathematical and definition-heavy, for a...