Solve Magnitude & Phase: 4 KHz Component in Periodic Pulse Train

[anik18]

magintude and phase...

Homework Statement

Hi Guys, I need to work it out how calculate magnitude and phase for below question. It is really complicated for me. So wait for your help. Below is a question and it figure is in attached file. Thankingyou

Calculate the magnitude and phase of the 4 KHz component in the spectrum of the periodic pulse train shown below. The pulse repetition rate is 1 KHz.

Homework Equations

The Attempt at a Solution

 

[Valkarie]

The magnitude of the 4 KHz component can be calculated using the Fourier transform. The Fourier transform of a periodic pulse train is a series of delta functions located at integer multiples of the repetition rate. This means that the 4 KHz component will have a magnitude equal to the amplitude of the pulse. The phase of the 4 KHz component can be calculated by summing the phases of the delta functions at 4 KHz and 1 KHz. The phase of the delta function at 4 KHz is 0, and the phase of the delta function at 1 KHz is equal to the phase of the pulse. Therefore, the phase of the 4 KHz component is equal to the phase of the pulse.

 

[piercas]

I would first start by breaking down the problem into smaller, more manageable parts. The first step would be to understand the concept of magnitude and phase in a periodic pulse train.

Magnitude refers to the amplitude or strength of a signal, while phase refers to the position of a signal in its cycle. In this case, we are interested in the 4 KHz component, which is a specific frequency within the periodic pulse train.

To calculate the magnitude and phase of the 4 KHz component, we can use Fourier analysis. This technique allows us to break down a complex signal into its individual frequency components. In this case, we are looking for the magnitude and phase of the 4 KHz component, which is a single frequency component.

To calculate the magnitude, we can use the formula:

Magnitude = amplitude of the signal at the specific frequency / amplitude of the fundamental frequency

In this case, the fundamental frequency is 1 KHz, and the amplitude of the signal at 4 KHz is approximately 0.5. Therefore, the magnitude of the 4 KHz component would be 0.5/1 = 0.5.

To calculate the phase, we can use the formula:

Phase = (time delay of the signal at the specific frequency / period) x 360 degrees

In this case, the time delay of the signal at 4 KHz is approximately 0.25 seconds, and the period is 1 second (since the pulse repetition rate is 1 KHz). Therefore, the phase of the 4 KHz component would be (0.25/1) x 360 = 90 degrees.

In summary, the magnitude of the 4 KHz component is 0.5 and the phase is 90 degrees. This means that the 4 KHz component has an amplitude that is half of the fundamental frequency and is delayed by 90 degrees compared to the fundamental frequency. These values can also be confirmed by looking at the attached figure, where we can see that the amplitude of the 4 KHz component is approximately half of the fundamental frequency and it is shifted by 90 degrees in the positive direction.