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By definition of fourier transform $$X(w)=\int_{-\infty}^\infty x(t) e^{-jwt} dt $$ Now what will happen to the answer of transform for example in case of $x(t)= cos(w_ot)$ if limit is 0 to A instead of $-\infty$ to $\infty$? For $x(t)=cos(w_ot)$ its fourier transform is given by $ X(w)= \pi[\delta(w-w_0) + \delta(w+w_o)]$ so if the limit is changed will it effect the answer? |
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Yes, it will affect the answer. What you're suggesting is known as the short-time Fourier transform. In the sinusoidal case that you proposed, you will observe spectral leakage, as the truncation of the integral limits is equivalent to multiplication of the sinusoid by a rectangular window function. This multiplication in the time domain maps to convolution in the frequency domain. The Fourier transform of a rectangular window is a sinc function, so the convolution will yield two sinc functions centered at the locations of the impulses in your original answer. |
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