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]

Continue reading## 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

Continue reading## Non-Data-Aided Carrier Phase Estimation

A carrier phase offset rotates the Rx constellation causing decision errors even in a perfectly noiseless environment. One of the techniques used to overcome this problem is to insert a known sequence at the start of the transmission known as a preamble. Then, the Rx can utilize these known symbols in the arriving signal to estimate the carrier phase and de-rotate the constellation. However, inserting a known sequence within the message decreases the spectral efficiency of the system. To avoid this cost, a phase estimator (as well as estimators for other distortions) can be derived in a non-data-aided fashion. One

Continue reading## Modulation Bandwidths

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 \begin{equation}\label{eqCommSystemBWPAM} BW_{\text{PAM}} = 0.5\left(1+\alpha\right)R_M \end{equation} QAM is basically a similar modulation scheme except that it is modulated on a carrier.

Continue reading## 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

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