Time domain view of sampling theorem

Sampling a Continuous-Time Signal

Most signals of our interest — wireless communication waveforms — are continuous-time as they have to travel through a real wireless channel. To process such a signal using digital signal processing techniques, the signal must be converted into a sequence of numbers. This can be done through the process of periodic sampling. From Continuous to Discrete Time Consider a band-limited continuous-time signal $s(t)$ and its frequency domain representation $S(F)$ with bandwidth $B$, shown in the above figure. A discrete-time signal $s[n]$ can be obtained by taking samples of $s(t)$ at equal intervals of $T_S$ seconds. This process is shown in

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A block diagram for the implementation of a digital filter and square timing recovery for L=4 samples/symbol

Digital Filter and Square Timing Recovery

We have seen before how a symbol timing offset severely impacts the constellation of the received symbols. Therefore, symbol timing recovery is one of the most crucial jobs of a digital communications receiver. In the days of analog clock recovery, a timing error detector provided the instant to sample the Rx waveform at 1 sample/symbol at the maximum eye opening. However, discrete-time processing opened the doors for better timing recovery schemes as an ever increasing number of transistors within the same area consistently keeps bringing the digital processing cost down. Consequently, the use of analog circuits to control the timing

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A matched filter

Convoluted Correlation between Matched Filter and Correlator

Now we turn out attention towards a topic that causes a lot of confusion for communications and DSP learners: what is the difference between a matched filter and a correlator in a communications receiver? Let us start with the definition of a correlator: A correlator is a device that performs correlation of a received signal with its template within a given window of time. In our context, that window of time is the symbol duration, $T_M$. So a correlator performs the following operations. It takes this sample-by-sample product and sums them together. Next, it samples the output of this accumulation

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What is a Signal?

In traditional Digital Signal Processing (DSP) courses, we usually learn about signals in a narrow context. While this is probably the correct approach for an introduction to this field, there is a broader theme to the concept of signals. Let us focus on the purely technical part first. A signal is any measurable quantity that varies with time (or some other independent variable). If you record the words you speak in a computer memory and plot against time, you get a speech signal. As an example, the figure below shows a signal for the words "Hello Wireless" recorded in a

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Illustration of peak ISI and asymmetry about -3dB point

Pulse Shaping Filter

In digital logic, a stream of 1s and 0s forms a sequence of rectangular pulses, which can be easily identified at the receiver side by a threshold. In time domain, everything looks nice and perfect. Let us investigate the system characteristics in frequency domain. In a Pulse Amplitude Modulation (PAM) system, the main component that defines the spectral contents of the signal is the pulse shape $p(nT_S)$ at the Tx. We start with our attention towards a simple rectangular pulse shape. Here is a brief outline of what we cover in this article. Table of Contents 1. Spectrum of a

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