## Modulation – From Numbers to Signals

The purpose of digital communications is to send digital data across a channel which can be wireless telephone lines coaxial cable optical fiber Ethernet USB chips on a printed circuit board Considering the examples shown in Figure above, clearly neither a bit sequence nor a symbol sequence can be transmitted on their own through these channels — as they are nothing more than a set of numbers. Therefore, a signal waveform is an appropriate tool that can travel down the channel and carry the required information — just like a train running on its track and carrying the load. For

## Transforming a Signal

Transforming a discrete-time signal — whether in time or amplitude — is certainly possible, and often in interesting ways. In practice, scaling and time shifting are the two most important signal modifications encountered. Scaling changes the values of dependent variable on amplitude-axis while time shifting affects the values of independent variable on time-axis. Below we describe addition and multiplication of two signals as well as scaling and time shifting a signal in detail. Addition For addition of two discrete-time signals, say $x[n]$ and $y[n]$, add the two signals sample-by-sample: $z[n] = x[n] + y[n]$ for every $n$, e.g., \begin{align*} z[0]

## DFT Examples

For understanding what follows, we need to refer to the Discrete Fourier Transform (DFT) and the effect of time shift in frequency domain first. Here, we discuss a few examples of DFTs of some basic signals that will help not only understand the Fourier transform but will also be useful in comprehending concepts discussed further. A Rectangular Signal A rectangular sequence, both in time and frequency domains, is by far the most important signal encountered in digital signal processing. One of the reasons is that any signal with a finite duration, say $T$ seconds, in time domain (that all practical

From the article on pulse shaping, we can correctly determine the occupied bandwidth for each modulation scheme where the Square-Root Raised Cosine spectrum shows the bandwidth of a Square-Root Raised Cosine pulse shape as $0.5(1+\alpha)R_M$. Also, we have discussed earlier that the spectrum approximately remains the same, provided that there is enough randomness in bit stream and the resulting symbols are equally likely and independent from each other. Therefore, the bandwidth for a PAM modulated signal can be given as $$\label{eqCommSystemBWPAM} BW_{\text{PAM}} = 0.5\left(1+\alpha\right)R_M$$ QAM is basically a similar modulation scheme except that it is modulated on a carrier.