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From Continuous Time to DiscreteTime

From Continuous Time to DiscreteTime

Learn how to acquire digital samples from continuous-time signals.

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Most signals in real life, e.g., heartbeat, brain waves, sound, music, wireless communication waveforms, etc., are continuous-time signals. To process such a signal using DSP techniques, it must be converted into a sequence of numbers. This can be done through the process of periodic sampling with the help of a device known as an analog-to-digital converter (ADC).

Time domain

Consider an analog signal x(t)x(t) as a function of time shown in the figure below. Then drawn in the figure is the spectrum of this signal.

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A signal and its spectrum
A signal and its spectrum

For digital signal processing, this signal needs to be converted into a series of samples, i.e., into a discrete-time version x[n]x[n]. This can be accomplished by sampling x(t)x(t) at regular intervals of TsT_s seconds. This process is mathematically represented as:

x[n]=x(t)t=nTS \begin{equation*} x[n] = x(t)\bigg| _{t=nT_S} \end{equation*}

The time interval denoted by TsT_s ...