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I have a square wave signal and its Fourier Series. This signal pass through an ideal low pass filter,H(f), which has a cutoff frequency of 4KHz. What is the resulting baseband bandwidth of the filtered signal?

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If you are passing the signal through an ideal low pass filter, then Fourier series coefficients of the filtered signal above $4$kHz vanish that is $a_k = 0 \forall k \ge \frac{4000T}{2\pi}$ and the rest of the coefficients remain same. Here $T$ is the period of the square wave.

Then what is the bandwidth? It depends on the definition of the bandwidth. Bandwidth usually means the the width of the continuous region in frequency domain within which most of the energy (usually $90\%$ or greater) of the signal is present. Bandwidth is usually inversely proportional to the width of the pulse of a rectangular pulse. Since its a square wave, the width is $\frac{T}{2}$ and hence the bandwidth is $\frac{2}{T}$. Now for the low pass filtered signal the bandwidth is $\min(\frac{2}{T},$4kHz$)$. $min$ is the minimum of operation.

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The baseband bandwidth is defined to be the highest frequency of the signal.

In our case, the baseband would be $f_{base} = (2k+1)f_{square}$, the largest frequency that is an odd multiple of the fundamental square wave frequency smaller than $4 kHz$.

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