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This is a statement I have read from a textbook:

Whenever we have periodic signals continuous or discrete time the frequency domain is discrete and time domain is continuous. Whenever we have aperiodic signals continuous or discrete time the frequency domain is continuous and time domain is discrete.

Are the above statements correct?

If yes, in case of a discrete time periodic signal how correct is it to say that it is continuous inthe time domain ? (The name itself suggests a discrete time periodic signal.)

In case of a continuous time aperiodic signal how correct is it to say that it is discrete in the time domain ? (The name itself suggests a continuous time aperiodic signal.)

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    $\begingroup$ These statements are either badly punctuated or have atrocious grammar. Any sane interpretations I can apply are self-contradictory (i.e. the first sentence starts by saying "whether continuous or discrete time" and ends by saying "time domain is continuous"), or plain wrong. Please edit your question to make sure your quote is accurate including punctuation, and please properly cite the book (title, author, publisher, date). $\endgroup$
    – TimWescott
    Commented Jul 7, 2021 at 2:54

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The key here is that

A signal that's discrete in one domain is periodic in the other (and vice versa)

The opposite holds as well: i.e. a signal that continuous in one domain is aperiodic in the other.

That gives a total of four possible types of signals and hence there are four different flavors of the Fourier Transform

  1. Time: continuous & aperiodic + Frequency: continuous & aperiodic -> Fourier Transform
  2. Time: continuous & periodic + Frequency: discrete & aperiodic -> Fourier Series
  3. Time: discrete & aperiodic + Frequency: continuous & periodic -> Discrete Time Fourier Transform
  4. Time: discrete & periodic + Frequency: discrete & periodic -> Discrete Fourier Transform

In case of discrete time periodic signal how much its correct to say that it is continuous in time domain ?

That's wrong (as least as I understand the sentence). The signal cannot be continuous and discrete in time at the same time. If it's periodic and discrete in time it will also be periodic and discrete in frequency (Case 4)

in case of continuous time aperiodic signal how much its correct to say that it is discrete in time domain ?

That's wrong too. It can't be continuous and discrete in time at the same time. These are opposites. If it's continuous and aperiodic in time, it's also continuous and aperiodic in frequency. (Case 1)

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