# Why do we use discrete-time sampling?

I have been reading Signals and Systems (2nd ed.) recently. Chapter 7 is about sampling. Oppenheim uses discrete-time signals and continuous-time signals to explain sampling.

• But why do we need discrete-time signals? We use the A/D to receive and convert the continuous-time signal, so the signal is already discrete. In other words, the discrete signals are the sampled version of continuous signals.
• Is there any practical meaning to sample discrete-time signals?
• Can you please clarify? You are asking three different questions and it's not clear what is causing your confusion. Also, please cite the book's definitions, for those of us who don't have immediate access to it. – MBaz Jul 22 '16 at 15:45

Sampling discrete-time signals, i.e., using only every $N^{th}$ sample of a sequence of samples, is useful for efficiently processing, transmitting, or storing information, if we can be sure that the sampling rate can be reduced without significant loss of information. This process is called decimation. Whether the sampling rate can be reduced or not depends on the frequency content of the signal.

Example: if the signal has a low pass characteristic with no significant components above $f_s/4$ (where $f_s$ is the sampling frequency), the sampling rate can be reduced by a factor of $2$, i.e., every other sample can be thrown away without losing information.

• Oh. You mean that decimation and interpolation are equal to sampling of discrete-time signals? – user22971 Jul 22 '16 at 12:21
• So , from continuous-time -> discrete-time we need impulse trains, this process is called A/D. Then we use interpolation or decimation to further process the signals. Is that right? – user22971 Jul 22 '16 at 12:23
• @user22971: You don't necessarily need to decimate or interpolate, but you can do it if it's useful. I wouldn't say that interpolation is sampling a discrete-time signal, but you can say that decimation is basically equivalent to sampling a discrete-time signal, i.e., taking samples of a (already sampled) signal. – Matt L. Jul 22 '16 at 12:30
• Oh. I am sorry , yep. Interpolation is adding points to the signal.. So A/D is necessary , but further processing with the sequences is not required. Is that right? – user22971 Jul 22 '16 at 12:59
• @user22971: The required processing depends on your application. – Matt L. Jul 22 '16 at 13:00

There is a difference between a quantized signal, i.e. taking only integer values, and a sampled signal, i.e. the value of which is only looked at on spaced instants.

A physical signal is usually continuous in values and in time.

An A/D converter turns a physical signal in a quantized one; if the converter is asynchronous, it generates a time-continuous signal.

Discrete-time/continuous-value signals are created by sample & hold circuits.

And digital signals processed by computers are usually both quantized and time-discrete.

unless it's a flash, the A/D needs time to figure out what the signal value is. what's it supposed to do while the continuous-time signal is changing as it is determining the signal value?

then with processing, how to do digital processing in a continuous-time context? totally parallel processing? lotsa gates. one of the promises of Digital Signal Processing over analog signal processing is that you need not change and rewire the hardware to change the algorithm. how're you gonna do that with a flash A/D and continuous-time processing?

• Er. Sorry, I do not get your point. I want to know whether$I \ need\ to\ sample \ the\ discrete-time\ signals$ , and if needed, why? – user22971 Jul 22 '16 at 11:57
• I think discrete-time signals is the result of sampling, why do I need to sample them again? – user22971 Jul 22 '16 at 11:59