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It's my first time studying DSP and I've faced a problem finding a convenient definition.

Are the following definitions correct? And if so why there are some resources defining it in other terms such as "Digital signal: is a signal with discrete time and discrete amplitude"

  1. Discrete time signal: X-axis (time) is discrete and Y-axis (amplitude) may be continuous or discrete.

  2. Continuous time signal: X-axis (time) is continuous and Y-axis (amplitude) may be continuous or discrete.

  3. Digital signal: Y-axis (amplitude) is discrete and X-axis (time) may be continuous or discrete.

  4. Analog signal: Y-axis (amplitude) is continuous and X-axis (time) may be continuous or discrete.

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  • $\begingroup$ that's pretty much correct. in fact, relatively few Analog Signal Processors are discrete time (CCD is the exception). and i don't know any continuous-time Digital signals except for the output of a flash A/D converter. $\endgroup$ – robert bristow-johnson Sep 27 '16 at 20:06
  • $\begingroup$ Any comparator might produce a continuous-time 1-bit Digital signal output. $\endgroup$ – hotpaw2 Sep 27 '16 at 20:29
  • $\begingroup$ @robertbristow-johnson I learned that "digital"=="time-discrete && value-discrete"! $\endgroup$ – Marcus Müller Sep 27 '16 at 23:20
  • $\begingroup$ @MarcusMüller i am not sure that it absolutely must be discrete time. but other than a flash A/D (which has a "thermometer code" output word and no clock), i dunno anything else digital that is not discrete time. as hotpaw points out, a comparator outputs a 1-bit digital signal and it is continuous time. a comparator is a 1-bit flash A/D. $\endgroup$ – robert bristow-johnson Sep 28 '16 at 13:46
  • $\begingroup$ gray encoders (position encoders utilizing gray code) are by design continuous-time and digital :-) $\endgroup$ – Olli Niemitalo Oct 7 '16 at 12:14
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A signal is indeed a function. Given a signal $f(x)$, according to whether continuous or discrete for both the variable $x$ and the function $f(x)$, there are four types of combinations:

(1) $\mathbf{continuous}$ $x$ and $\mathbf{continuous}$ $f(x)$

This is the most common $\mathbf{analog}$ signal.

(2) $\mathbf{continuous}$ $x$ and $\mathbf{discrete}$ $f(x)$

For this one, we can imagine the ideal base-band waveform used in digital communication. As this one: enter image description here

(3) $\mathbf{discrete}$ $x$ and $\mathbf{continuous}$ $f(x)$

This is indeed the signal in most the "digital signal processing" textbooks. An example, as others have pointed out, is the output of the CCD sensor.

(4) $\mathbf{discrete}$ $x$ and $\mathbf{discrete}$ $f(x)$

This is the $\mathbf{digital}$ signal. Digital signals are used in practical implementation aspects, and they actually exist in a conceptual manner.

If we concern the discrete feature of the function, the problem will be more complex, therefore, in most "digital signal processing" textbooks, the signals are $\mathbf{not}$ digital indeed. An interesting fact is that, for the the classical textbook by A. V. Oppenheim, the name was "digital signal processing" in the 1st edition, but the name was changed to "discrete-time signal processing" for the later editions.

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There are 2 types of digital here.

Think of OFDM using 16QAM. You have a bit stream that you are taking and correlating it to discrete amplitudes of 2 4PAM signals, one in phase and one in quadrature. This is a digital signal right, due to the discrete amplitude part. But how do you transmit it? With a digital sample to analogue converter on the output of the DFT. You are transmitting a digital signal (superposition of 2 x 4PAM x no. usable subcarriers) but it is being transmitted in an analogue manner. Analogue in this sense means a physical signal on a physical medium, where digital in this sense means mathematical data stored in memory.

The other sense of analogue and digital is the discrete amplitude part; the notion that each amplitude represents data individually (either raw data or a bit stream as a result of discretisation and quantisation of an analogue signal) and that the waveform consists of a string of symbols from a set of discrete symbols forms. These signals are often shown with continuous time. i.e. a square wave, which is clearly continuous

$$\sum_{ \substack {k\in \mathbb{N}_1\text{\2} \mathbb{N}_1\\ (k-1)|4}} \frac{1}{k} \cos(\omega_k t) - \sum_{ \substack {k\in \mathbb{N}_1\text{\2} \mathbb{N}_1\\ (k-3)|4}}\frac{1}{k} \cos(\omega_k t)$$

So analogue in this sense means continuous amplitude whereas digital means discrete amplitude. A digital or analogue signal with discrete time however is essentially the first definition of digital. Discrete time only exists in memory and continuous time only exists on the physical medium.

The dichotomy is between analogue and digital domain in the first paragraph and analogue and digital signal in the second paragraph. Analogue domain means continuous time and analogue signal means continuous amplitude.

Digital modulation is the modulating of a signal within memory i.e. the digital domain. The output is typically a digital signal of discrete symbols but it can also be an analogue signal, the signal is then converted to the analogue domain.

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