Spectrum of windowed version of original continuous signal - Signal Processing Stack Exchange most recent 30 from dsp.stackexchange.com 2019-09-19T03:17:53Z https://dsp.stackexchange.com/feeds/question/37548 https://creativecommons.org/licenses/by-sa/4.0/rdf https://dsp.stackexchange.com/q/37548 0 Spectrum of windowed version of original continuous signal Mahdi https://dsp.stackexchange.com/users/24932 2017-02-11T23:55:13Z 2017-02-14T07:47:57Z <p>Suppose we have the complex signal $x(t)= \exp(j\omega_0 t)$. Using the properties of Fourier transform we can prove its CTFT is Dirac $\delta$ function. </p> <p>If any one ask me about the spectrum of $x(t)$ "Does $x(t)$ has continuous spectrum or discrete spectrum", my answer will be "The spectrum of $x(t)$ is discrete".</p> <p>Now, if I apply a rectangular window to the complex exponential $x(t)$ in the time domain and then take CTFT I will end up with $\mathrm{sinc}$ function. Now the spectrum of the windowed complex exponential is continuous and not discrete.Is this interpretation true? </p> https://dsp.stackexchange.com/questions/37548/-/37553#37553 3 Answer by hops for Spectrum of windowed version of original continuous signal hops https://dsp.stackexchange.com/users/25664 2017-02-12T08:20:41Z 2017-02-13T21:02:59Z <p>If you don't understand the difference between the <a href="https://en.wikipedia.org/wiki/Fourier_transform" rel="nofollow noreferrer">Continuous Time Fourier Transform</a> (CTFT), the <a href="https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" rel="nofollow noreferrer">Discrete Time Fourier Transform</a> (DTFT) and the <a href="https://en.wikipedia.org/wiki/Discrete_Fourier_transform" rel="nofollow noreferrer">Discrete Fourier Transform</a> (DFT), now would be a good time to read about them. The <em>very short</em> version is that the DTFT yields the (continuous-valued) spectrum of a sequence (i.e., a sampled signal). The DFT computation results in a sampled version of the DTFT. To apply the DFT requires a finite number of samples (i.e., a time-domain window) whereas such restrictions are not placed on the DTFT in general. On the other hand, the CTFT deals with continuous time signals. There is a lot more to all of this, and I recommend you read more. </p> <p>It is true that a complex-valued signal $x(t) = \exp\left(j \omega_0 t\right)$ and the Dirac delta function form a CTFT pair. I can agree with you that the spectrum of this signal is discrete (nonzero for a finite number of frequencies, in this case a single frequency).</p> <p>After applying a rectangular window in time to the complex-valued signal $x(t)$, the CTFT is a frequency-translated sinc function centered at $\omega_0$ with a lobe width inversely proportional to the size of the time domain window. This shows us that the result of truncating a time domain signal with infinite support in time and a discrete spectrum in frequency can lead to a new time domain signal with finite support and a continuous spectrum. So, the answer is <em>yes</em>, the interpretation given in the question is true.</p>