Main Lobe Peak: Width and Zero Crossings

The DFT of a rectangular signal $x[n]$ is found to be a sinc signal.

$$X[k]=\frac{\sin\left( \pi\frac{k}{N}L\right)}{\pi\frac{k}{N}}$$

This sinc signal has a main lobeA lobe is the part of the spectrum that looks like an inverted parabola. that is centered around the frequency bin $k = 0$. To find the amplitude at the y-axis that determines the height of the main lobe, we can’t put $k=0$ in the expression above because both the numerator and denominator become zero, generating an indeterminate form.

Main lobe peak

From the DFT definition, we have

$$X[k]=\sum_{n=-\frac{N}{2}}^{\frac{N}{2}}x[n]e^{-j2\pi\frac{k}{N}n}=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{-j2\pi\frac{k}{N}n}$$

for an odd-length $L$ rectangular signal. Plug in $k=0$ at this stage to get:

$$X[0]=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}1=L$$

Interestingly, the value $X[0]$ is the sum of all the time domain samples and is known as its DC value if divided by $N$.