Mathematics - Signals And Systems - Processing Continuous-Time Signals as Discrete-Time Signals



Hey it's a me again @drifter1!

Today we continue with my mathematics series about Signals and Systems in order to cover Processing Continuous-Time Signals as Discrete-Time Signals.

So, without further ado, let's get straight into it!

Continuous-Time to Discrete-Time Conversion (and vise versa)

Maybe the most important application of sampling is its role in processing continuous-time signals using discrete-time systems. Continuous-Time signals have to be band-limited in order for the conversion to work. If they are already band-limited they can easily be sampled, and those samples are then converted to some kind of discrete-time representation. On the other hand, if they are not band-limited that are first processed with an anti-aliasing filter (which band-limits them) and then sampled.

The resulting discrete-time representation of the continuous-time signals can be processed by discrete-time systems and when the whole procedure is finished desampled/reconstructed into continuous-time signals. This is done using preferably band-limited interpolation, which allows converting the sequence to an (im)pulse train. This (im)pulse train is then processed through a low-pass filter, which in turn gives the final continuous-time signal. Of course the exact sampling period, the periodicy of the to-be-sampled signal, and much much more that we already discussed throughout the series, also affects the outcome of this whole procedure.

Discrete-Time Processing

Whilst converting the impulse train of samples to a sequence of samples, we basically normalize the time-axis. Recalling the properties of the Fourier Transform, its easy to conclude that the discrete-time Fourier transform of the sequence of samples gives the same result as the continuous-time Fourier transform of the impulse train, if the frequency-axis is normalized in that case. In other words, processing the continuous-time signal as an discrete-time signal is not violating the Fourier transform principles, and the result after the sampler is the same that we would get processing the continuous-time signal directly.

To get even more specific, while converting the result back to a continuous-time signal, we basically "un-normalize" the frequency-axis. So, the overall system which processes the discrete-time representation is equivalent to a continuous-time filter with a frequency response that is the same as the frequency resposne of the discrete-tiem filter if linear scaled on the frequency-axis. The cut-off frequency in this case depends on the sampling frequency, and so it can be varied by varying the sampling frequency.

Complete Block Diagram



  1. Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.


Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using and GeoGebra

Previous articles of the series


LTI Systems and Convolution

Fourier Series and Transform

Filtering, Sampling, Modulation, Interpolation

  • Filtering → Convolution Property, Ideal Filters, Series R-C Circuit and Moving Average Filter Approximations
  • Continuous-Time Modulation → Getting into Modulation, AM and FM, Demodulation
  • Discrete-Time Modulation → Applications, Carriers, Modulation/Demodulation, Time-Division Multiplexing
  • Sampling → Sampling Theorem, Sampling, Reconstruction and Aliasing
  • Interpolation → Reconstruction Procedure, Interpolation (Band-limited, Zero-order hold, First-order hold)

Final words | Next up

And this is actually it for today's post!

Next time we will get into discrete-time sampling...

See Ya!

Keep on drifting!

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