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I'm just wondering about the CTFT. My lecturer refers to capital Omega $\Omega$ in the following as being the continuous-time frequency:

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Why is it called continuous-time frequency here but in the DTFT the small omega is called discrete-time frequency? CTFT and DTFT both give continuous signals out.

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    $\begingroup$ they are both continuous spectra, but the signals that each spectrum is representing are different. one is continuous-time, $x(t)$, and the other is discrete-time $x[n] \triangleq x(nT)$. $\endgroup$ – robert bristow-johnson Oct 2 '18 at 23:59
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$\Omega$ is the usual angular frequency in radians per second, and is equal to $2 \pi f$. It is the way to measure frequency for continuous-time signals.

In discrete-time, frequency is measured in radians per sample, and is denoted as $\omega$. Here, a frequency component with $\omega = 2\pi$ is indistinguishable from $\omega = 0$.

$\Omega$ and $\omega$ is the notation in texts such as Oppenheim-Schafer. Other authors use $\theta$ for discrete-time, and the classic $\omega$ for continuous-time signals.

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  • $\begingroup$ Ahhh. This makes soo much more sense. $\endgroup$ – AlfroJang80 Oct 3 '18 at 0:12
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I believe there is an issue in the interpretation of the F in CTFT. FT stands for Fourier Transform. Each of the dual variables time and frequency can be either continuous or discrete. The Continuous-Time Fourier Transform (CTFT) is often simply called "Fourier Transform", and uses both continuous time and frequency. For the Discrete-Time Fourier Transform (DTFT), time is discrete, yet the frequency is continuous (with inherent periodicity).

Other avatars are Fourier series (continuous in time, discrete in frequency) and the Discrete Fourier Transform (DFT), discrete in both domains. All of the above have subtle relationships, and share a handful of similar properties. In fact, CTFT is not piratically usable for computations, and DFT (and FFT) are best efforts to turn CTFT into something usable.

So I am not sure that "continuous-time frequency" is meaningful by itself. It might be a short-hand for "continuous in time (and) frequency".

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