Multirate Signal Processing Concepts in Digital Communications
Multirate systems are building blocks commonly used in digital signal processing (DSP). Their function is to alter the rate of the discrete-time signals, by adding or deleting a portion of the signal samples. They are essential in various standard signal processing techniques such as signal analysis, denoising, compression and so forth. During the last decade, however, they have increasingly found applications in new and emerging areas of signal processing, as well as in several neighboring disciplines such as digital communications. The main contribution of this thesis is aimed towards a better understanding of multirate systems and their use in modern communication systems. To this end, we first study a property of linear systems appearing in certain multirate structures. This property is called biorthogonal partnership and represents a terminology introduced recently to address a need for a descriptive term for such class of filters. In the thesis we especially focus on the extensions of this simple idea to the case of vector signals (MIMO biorthogonal partners) and to accommodate for nonintegral decimation ratios (fractional biorthogonal partners). The main results developed here study the properties of biorthogonal partners, e.g., the conditions for the existence of stable and of finite impulse response (FIR) partners. In this context we develop the parameterization of FIR solutions, which makes the search for the best partner in a given application analytically tractable. This proves very useful in their central application, namely, channel equalization in digital communications with signal oversampling at the receiver. A good channel equalizer in this context is one that helps neutralize the distortion on the signal introduced by the channel propagation but not at the expense of amplifying the channel noise. In the second part of the thesis, we focus on another class of multirate systems, used at the transmitter side in order to introduce redundancy in the data stream. This redundancy generally serves to facilitate the equalization process by forcing certain structure on the transmitted signal. We first consider the transmission systems that introduce the redundancy in the form of a cyclic prefix. The examples of such systems include the discrete multitone (DMT) and the orthogonal frequency division multiplexing (OFDM) systems. We study the signal precoding in such systems, aimed at improving the performance by minimizing the noise power at the receiver. We also consider a different class of communication systems with signal redundancy, namely, the multiuser systems based on code division multiple access (CDMA). We specifically focus on the special class of CDMA systems called `a mutually orthogonal usercode receiver' (AMOUR). We show how to find the best equalizer from the class of zero-forcing solutions in such systems, and then increase the size of this class by employing alternative sampling strategies at the receiver.
Fractional Delay Farrow Filter
The Fractional Delay Farrow Filter is a digital filter that delays the discrete-time input signal by a fraction of the sample period. There are many applications where such a delay is necessary. As an example one can consider symbol synchronization in digital receivers, conversion between arbitrary sampling frequencies, echo cancellation, speech coding and speech synthesis, modeling of musical instruments, etc.
Generating Complex Baseband and Analytic Bandpass Signals
There are so many different time- and frequency-domain methods for generating complex baseband and analytic bandpass signals that I had trouble keeping those techniques straight in my mind. Thus, for my own benefit, I created a kind of reference table showing those methods. I present that table for your viewing pleasure in this document.
The DFT Magnitude of a Real-valued Cosine Sequence
This article may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.
Sum of Two Equal-Frequency Sinusoids
The sum of two equal-frequency real sinusoids is itself a single real sinusoid. However, the exact equations for all the various forms of that single equivalent sinusoid are difficult to find in the signal processing literature. Here we provide those equations.
Digital Filtering in the Frequency Domain
Time domain digital filtering, whether implemented using FIR or IIR techniques, has been very well documented in literature and been thoroughly used in many base band processor designs. However, with the advent of software defined radios as well as CPU support in more recent baseband processors, it has become possible and often desirable to filter signals in software rather than digital hardware. Whereas, time domain digital filtering can certainly be implemented in software as well, it becomes highly inefficient as the number of filter taps grows. Frequency domain filtering, using FFT and IFFT operations, is significantly more efficient and surprisingly easy to understand. This document introduces the reader to frequency domain filtering both in theory and in practice via a MatLab script.
The DFT of Finite-Length Time-Reversed Sequences
Recently I've been reading papers on underwater acoustic communications systems and this caused me to investigate the frequency-domain effects of time-reversal of time-domain sequences. I created this article because there is so little coverage of this topic in the literature of DSP.
Correcting an Important Goertzel Filter Misconception
Correcting an Important Goertzel Filter Misconception
Understanding the 'Phasing Method' of Single Sideband Demodulation
There are four ways to demodulate a transmitted single sideband (SSB) signal. Those four methods are: synchronous detection, phasing method, Weaver method, and filtering method. Here we review synchronous detection in preparation for explaining, in detail, how the phasing method works. This blog contains lots of preliminary information, so if you're already familiar with SSB signals you might want to scroll down to the 'SSB DEMODULATION BY SYNCHRONOUS DETECTION' section.
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
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
Multirate Systems and Filter Banks
During the last two decades, multirate filter banks have found various applications in many different areas, such as speech coding, scrambling, adaptive signal processing, image compression, signal and image processing applications as well as transmission of several signals through the same channel. The main idea of using multirate filter banks is the ability of the system to separate in the frequency domain the signal under consideration into two or more signals or to compose two or more different signals into a single signal.
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.
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.
The Swiss Army Knife of Digital Networks
This article describes a general discrete-signal network that appears, in various forms, inside so many DSP applications.
Digital PLL's -- Part 1
We will use Matlab to model the DPLL in the time and frequency domains (Simulink is also a good tool for modeling a DPLL in the time domain). Part 1 discusses the time domain model; the frequency domain model will be covered in Part 2. The frequency domain model will allow us to calculate the loop filter parameters to give the desired bandwidth and damping, but it is a linear model and cannot predict acquisition behavior. The time domain model can be made almost identical to the gate-level system, and as such, is able to model acquisition.
Digital Envelope Detection: The Good, the Bad, and the Ugly
Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place. Here I focus of envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope) contain some sort of information. Let's begin by looking at the simplest envelope detection method.
Lecture Notes on Elliptic Filter Design
Elliptic filters, also known as Cauer or Zolotarev filters, achieve the smallest filter order for the same specifications, or, the narrowest transition width for the same filter order, as compared to other filter types. On the negative side, they have the most nonlinear phase response over their passband. In these notes, we are primarily concerned with elliptic filters. But we will also discuss briefly the design of Butterworth, Chebyshev-1, and Chebyshev-2 filters and present a unified method of designing all cases. We also discuss the design of digital IIR filters using the bilinear transformation method.
The DFT Magnitude of a Real-valued Cosine Sequence
This article may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.
Sum of Two Equal-Frequency Sinusoids
The sum of two equal-frequency real sinusoids is itself a single real sinusoid. However, the exact equations for all the various forms of that single equivalent sinusoid are difficult to find in the signal processing literature. Here we provide those equations.
Fractional Delay Farrow Filter
The Fractional Delay Farrow Filter is a digital filter that delays the discrete-time input signal by a fraction of the sample period. There are many applications where such a delay is necessary. As an example one can consider symbol synchronization in digital receivers, conversion between arbitrary sampling frequencies, echo cancellation, speech coding and speech synthesis, modeling of musical instruments, etc.
The Swiss Army Knife of Digital Networks
This article describes a general discrete-signal network that appears, in various forms, inside so many DSP applications.
A Wide-Notch Comb Filter
This article describes a linear-phase comb filter having wider stopband notches than a traditional comb filter.
An Introduction To Compressive Sampling
This article surveys the theory of compressive sensing, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition.
Zero-Order-Hold function as a model for DAConverters
The output of a digital to analog converter, short DAC, is a constant analog signal between two discrete samples. The DAC's output register retains its value from one sample up to the next sample. The network, which converts the binary value of the register into an analog voltage thus supplies a constant voltage and that leads to a stepped output signal. The analog smoothing filter connected at the output together with the frequency response of the DAC, modeled with a Zero-Order-Hold function, results in distortions. These are examined here and solutions to compensate for them are presented.
Multirate Systems and Filter Banks
During the last two decades, multirate filter banks have found various applications in many different areas, such as speech coding, scrambling, adaptive signal processing, image compression, signal and image processing applications as well as transmission of several signals through the same channel. The main idea of using multirate filter banks is the ability of the system to separate in the frequency domain the signal under consideration into two or more signals or to compose two or more different signals into a single signal.
Digital Filtering in the Frequency Domain
Time domain digital filtering, whether implemented using FIR or IIR techniques, has been very well documented in literature and been thoroughly used in many base band processor designs. However, with the advent of software defined radios as well as CPU support in more recent baseband processors, it has become possible and often desirable to filter signals in software rather than digital hardware. Whereas, time domain digital filtering can certainly be implemented in software as well, it becomes highly inefficient as the number of filter taps grows. Frequency domain filtering, using FFT and IFFT operations, is significantly more efficient and surprisingly easy to understand. This document introduces the reader to frequency domain filtering both in theory and in practice via a MatLab script.
Implementation of a Tx/Rx OFDM System in a FPGA
The aim of this project consists in the FPGA design and implementation of a transmitter and receiver (Tx/Rx) multicarrier system such the Orthogonal Frequency Division Multiplexing (OFDM). This Tx/Rx OFDM subsystem is capable to deal with with different M-QAM modulations and is implemented in a digital signal processor (DSP-FPGA). The implementation of the Tx/Rx subsystem has been carried out in a FPGA using both System Generator visual programming running over Matlab/Simulink, and the Xilinx ISE program which uses VHDL language. This project is divided into four chapters, each one with a concrete objective. The first chapter is a brief introduction to the digital signal processor used, a field-programmable gate array (FPGA), and to the VHDL programming language. The second chapter is an overview on OFDM, its main advantages and disadvantages in front of previous systems, and a brief description of the different blocks composing the OFDM system. Chapter three provides the implementation details for each of these blocks, and also there is a brief explanation on the theory behind each of the OFDM blocks to provide a better comprehension on its implementation. The fourth chapter is focused, on the one hand, in showing the results of the Matlab/Simulink simulations for the different simulation schemes used and, on the other hand, to show the experimental results obtained using the FPGA to generate the OFDM signal at baseband and then upconverted at the frequency of 3,5 GHz. Finally the conclusions regarding the whole Tx/Rx design and implementation of the OFDM subsystem are given.
Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals
In digital signal processing (DSP) we're all familiar with the processes of bandpass sampling an analog bandpass signal and downsampling a digital bandpass signal. The overall spectral behavior of those operations are well-documented. However, mathematical expressions for computing the translated frequency of individual spectral components, after bandpass sampling or downsampling, are not available in the standard DSP textbooks. This document explains how to compute the frequencies of translated spectral components and provide the desired equations in the hope that they are of use to you.
Digital PLL's -- Part 1
We will use Matlab to model the DPLL in the time and frequency domains (Simulink is also a good tool for modeling a DPLL in the time domain). Part 1 discusses the time domain model; the frequency domain model will be covered in Part 2. The frequency domain model will allow us to calculate the loop filter parameters to give the desired bandwidth and damping, but it is a linear model and cannot predict acquisition behavior. The time domain model can be made almost identical to the gate-level system, and as such, is able to model acquisition.
