Setting Carrier to Noise Ratio in Simulations
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/NThe 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
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:
$$y=...
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...
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...
Model Signal Impairments at Complex Baseband
In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise. To provide concrete examples, we’ll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code. Although our example system is QAM, the models can be used for any complex-baseband signal.
I used a very simple complex-baseband model of a QAM system in my last
Compute Modulation Error Ratio (MER) for QAM
This post defines the Modulation Error Ratio (MER) for QAM signals, and shows how to compute it. As we’ll see, in the absence of impairments other than noise, the MER tracks the signal’s Carrier-to-Noise Ratio (over a limited range). A Matlab script at the end of the PDF version of this post computes MER for a simplified QAM-64 system.
Figure 1 is a simplified block diagram of a QAM system. The transmitter includes a source of QAM symbols, a root-Nyquist...
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...
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...
A Two Bin Solution
Introduction
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showing an implementation of how the parameters of a real pure tone can be calculated from just two DFT bin values. The equations from previous articles are used in tandem to first calculate the frequency, and then calculate the amplitude and phase of the tone. The approach works best when the tone is between the two DFT bins in terms of frequency.
The Coding...IIR Bandpass Filters Using Cascaded Biquads
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...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...
The Exponential Nature of the Complex Unit Circle
IntroductionThis is an article to hopefully give an understanding to Euler's magnificent equation:
$$ e^{i\theta} = cos( \theta ) + i \cdot sin( \theta ) $$
This equation is usually proved using the Taylor series expansion for the given functions, but this approach fails to give an understanding to the equation and the ramification for the behavior of complex numbers. Instead an intuitive approach is taken that culminates in a graphical understanding of the equation.
Complex...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...
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...
IIR Bandpass Filters Using Cascaded Biquads
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...
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...
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...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...
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.
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...
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...
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...
IIR Bandpass Filters Using Cascaded Biquads
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...
Sinusoidal Frequency Estimation Based on Time-Domain Samples
The topic of estimating a noise-free real or complex sinusoid's frequency, based on fast Fourier transform (FFT) samples, has been presented in recent blogs here on dsprelated.com. For completeness, it's worth knowing that simple frequency estimation algorithms exist that do not require FFTs to be performed . Below I present three frequency estimation algorithms that use time-domain samples, and illustrate a very important principle regarding so called "exact"...
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...
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.
Design a DAC sinx/x Corrector
This post provides a Matlab function that designs linear-phase FIR sinx/x correctors. It includes a table of fixed-point sinx/x corrector coefficients for different DAC frequency ranges.
A sinx/x corrector is a digital (or analog) filter used to compensate for the sinx/x roll-off inherent in the digital to analog conversion process. In DSP math, we treat the digital signal applied to the DAC is a sequence of impulses. These are converted by the DAC into contiguous pulses...
