Which of the following represent impulse response of an ideal LPF ?

Explanation:

Impulse Response of an Ideal Low Pass Filter (LPF)

Definition: The impulse response of a system is the output of the system when an impulse signal is applied as input. For an ideal Low Pass Filter (LPF), the frequency response is a rectangular function in the frequency domain, which allows frequencies within a specified range (passband) to pass through and attenuates frequencies outside this range (stopband).

Mathematical Representation:

An ideal low-pass filter has the following frequency response:

H(f) = 1 for |f| ≤ f_c

H(f) = 0 for |f| > f_c

Where:

• H(f): Frequency response of the filter
• f: Frequency variable
• f_c: Cutoff frequency

The impulse response, h(t), of the filter is obtained by taking the inverse Fourier Transform of its frequency response:

h(t) = ∫_-∞^∞ H(f) × e^j2πft df

For the ideal LPF, substituting the value of H(f):

h(t) = ∫_-fc^f_c e^j2πft df

Solving this integral gives:

h(t) = 2f_c × sinc(2f_ct)

Where the sinc function is defined as:

sinc(x) = sin(πx) / (πx)

Key Characteristics:

• The impulse response of an ideal LPF is a sinc function, which extends infinitely in both directions in time.
• The sinc function exhibits oscillatory behavior, with the main lobe centered at t = 0 and side lobes that decrease in amplitude as |t| increases.
• The main lobe width is inversely proportional to the cutoff frequency f_c.

Correct Option Analysis:

The correct option is:

Option 1: The impulse response of an ideal LPF is represented by the sinc function.

This option correctly identifies that the impulse response of an ideal low-pass filter is a sinc function, as derived mathematically above