A rectangular signal and its upsampled version in time and frequency domains

Sample Rate Conversion

In the discussion on sampling, the process of sampling a continuous-time signal was discussed in detail and subsequently sampling theorem was derived. In many applications, resampling an already digitized signal is mandatory for an efficient system design. In wireless communications, sample rate conversion is utilized for upconversion and downconversion to a desired frequency, filtering stages in the digital frontend and sometimes for carrier and timing synchronization during signal acquisition. See the Cascade Integrator Comb (CIC) filters for how to accomplish this task with minimal resources. In discrete domain, sample rate can be reduced by discarding intermediate samples periodically called downsampling

Continue reading
A Phase Locked Loop (PLL) for digital symbol timing recovery

Phase Locked Loop (PLL) for Symbol Timing Recovery

A Phase Locked Loop (PLL) is a device used to synchronize a periodic waveform with a reference periodic waveform. It is an automatic control system in which the phase of the output signal is locked to the phase of the input reference signal. In the context of carrier phase synchronization, we talk about tracking the phase of an input reference sinusoid. For carrier frequency synchronization, a Frequency Locked Loop (FLL) is implemented. For the purpose of timing synchronization, the target is to adjust the timing phase of a receiver clock to that of the transmitter clock such that one sample/symbol

Continue reading
A signal broken down into scaled and shifted impulses


Understanding convolution is the biggest test DSP learners face. After knowing about what a system is, its types and its impulse response, one wonders if there is any method through which an output signal of a system can be determined for a given input signal. Convolution is the answer to that question, provided that the system is linear and time-invariant (LTI). We start with real signals and LTI systems with real impulse responses. The case of complex signals and systems will be discussed later. Convolution of Real Signals Assume that we have an arbitrary signal $s[n]$. Then, $s[n]$ can be

Continue reading
A channel with 8 taps demonstrating the main cursor, precursor ISI and postcursor ISI

How Decision Feedback Equalizers (DFE) Work

We started the classification of equalization algorithms by introducing the need for equalization in wireless communication systems. We said that the wireless channel is a source of severe distortion in the received (Rx) signal and our main task is to remove the resulting Inter-Symbol Interference (ISI) from the Rx samples. Equalization refers to any signal processing technique in general and filtering in particular that is designed to eliminate or reduce this ISI before symbol detection. In essence, the output of an equalizer should be a Nyquist pulse for a single symbol case. A conceptual block diagram of the equalization process

Continue reading
A prism decomposes the white light into 7 colours

The Discrete Fourier Transform (DFT)

Learned in some other articles on this website, the following three important concepts take us to the core of the Discrete Fourier Transform (DFT) idea. Regardless of the signal shape, most signals of practical interest can be considered as a sum of complex sinusoids oscillating at different frequencies. A set of $N$ orthogonal complex sinusoids can be constructed within a span of $N$ time domain samples. Each `tick’ or bin on the discrete frequency axis denotes the discrete frequency $k/N$ of such a complex sinusoid. To understand how a set of sinusoids with $N$ discrete frequencies can sum up to

Continue reading