## 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 computer. Some familiar signals from everyday life are temperature or pressure readings, GPS signals that determine your vehicle’s position, biological

## 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 Rectangular Pulse 2. Reducing the bandwidth 3. Coefficients of an Ideal Pulse Auto-correlation 4. Raised Cosine (RC) Filter 5. Resulting

## Computing Error Rates

Having built a simple digital communication system, it is necessary to know how to measure its performance. As the names say, Symbol Error Rate (SER) and Bit Error Rate (BER) are the probabilities of receiving a symbol and bit in error, respectively. SER and BER can be approximated through simulating a complete digital communication system involving a large number of bits and comparing the ratio of symbols or bits received in error to the total number of bits. Hence, $$\label{eqCommSystemSER} \text{SER} = \frac{\text{No. of symbols in error}}{\text{Total no. of transmitted symbols}}$$ and $$\label{eqCommSystemBER} \text{BER} = \frac{\text{No. of bits in error}}{\text{Total no. of transmitted bits}}$$ Till now, the only imperfection between Tx and Rx entities we have covered is

Although complex notation is not complex to understand, I attempt to avoid complex notation altogether while writing DSP articles. If you are interested in where these complex numbers come from, you can read my real-imaginative guide to complex numbers. A complex number is defined as an ordered pair of real numbers in $(x,y)$-plane. In that respect, complex numbers can be considered as vectors with initial point on the origin $(0,0)$. Addition of complex numbers is then similar to the addition of vectors in $(x,y)$-plane from this perspective. However, multiplication is well defined for complex numbers while it is not defined for vectors — the dot product of two vectors is a scalar, not a vector, while the cross product of