Question

The signal f(t) = 10⁴ rect (10⁴ t) is applied at the input of an ideal low-pass filter with a frequency response H (ω)= rect (ω/40,000), where rect (x/d) is the rectangular given by

rect (x/d) = {1 ︱x︱< d/2
{0︱x︱> d/2
(i) sketch F (ω)
(ii) sketch H (ω)
(iii) sketch Y (ω)

Answer 1

The student's question involves sketching the frequency response of a signal, the response of an ideal low-pass filter, and the output signal after filtering. It requires an understanding of Fourier transforms, sinc functions, and the convolution theorem in signal processing.

Step-by-step explanation:

The student's question asks about signal processing and involves applying a signal to an ideal low-pass filter. The specific tasks are to sketch the frequency representation of a signal, the frequency response of the filter, and the output signal in the frequency domain.

Sketching F (ω)

The Fourier transform of the signal f(t) = 10⁴ rect(10⁴ t) will result in a sinc function centered at the origin, with its main lobe having a width determined by the inverse of the time domain signal's width.

Sketching H (ω)

The frequency response function H (ω) = rect(ω/40,000) will be a rectangle centered at the origin with a width of 40,000 radians per second, indicating the bandwidth of the filter.

Sketching Y (ω)

To obtain the output signal Y (ω), one would convolve the input signal frequency spectrum F (ω) with the filter's frequency response H (ω). This is equivalent to multiplying F (ω) and H (ω) in the frequency domain. The resulting spectrum will show which frequencies are passing through the filter and which are attenuated.