The purpose of this article is to summarize some useful DFT properties in a table. My favorite property is the beautiful symmetry depicted by continuous and discrete Fourier transforms. However, if you feel that this particular content is not as descriptive as the other posts on this website are, you are right. As opposed to the rest of the content on the website, we do not intend to derive all the properties here. Instead, based on what we have learned, some important properties of the DFT are summarized in the table below with an expectation that the reader can derive

Continue reading## Time-Invariant Systems

A system is time-invariant if shifting the input sequence on time axis leads to an equivalent shift of the output sequence along the time axis, with no other changes.

Continue reading## Demodulation – From Signals Back to Numbers

Remember that in the article on correlation, we discussed that correlation of a signal with proper normalization is maximum with itself and lesser for all other signals. Since the number of possible signals is limited in a digital communication system, we can use the correlation between incoming signal $r(nT_S)$ and possible choices $s_0(nT_S)$ and $s_1(nT_S)$ in a digital receiver. Consequently, a decision can be made in favor of the one with higher correlation. It turns out that the theory of maximum likelihood detection formalizes this conclusion that it is the optimum receiver in terms of minimizing the probability of error.

Continue reading## 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

Continue reading## 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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