Trigonometric function and complex exponential

It is important for the convolution result.In summary, the problem is regarding the multiplication of a trigonometric function and a complex exponential. The solution involves using the equation cos(ωt) = 1/2ejωt + 1/2e-jωt and results in Ak/2 + (Ak/2)e-j2ωt. To convolute with a real function, the imaginary part should not be ignored as it is important for the convolution result.
  • #1
asifadio
10
1
1. Homework Statement
- multiplication of trigonometric function and complex exponential



2. Homework Equations
the question is, Akcos(ωt) × e-jωt



3. The Attempt at a Solution
it is, Ak/2 + (Ak/2)e-j2ωt ?
by using cos(ωt) = 1/2ejωt + 1/2e-jωt
 
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  • #2
asifadio said:
1. Homework Statement
- multiplication of trigonometric function and complex exponential



2. Homework Equations
the question is, Akcos(ωt) × e-jωt



3. The Attempt at a Solution
it is, Ak/2 + (Ak/2)e-j2ωt ?
by using cos(ωt) = 1/2ejωt + 1/2e-jωt

Looks ok to me.
 
  • #3
thanks!
but if i want to convolute with some real function, f(x),
can i just take a real part of
Ak/2 + (Ak/2)e-j2ωt

which is Ak/2 + (Ak/2)(2cos(ωt))?
 
  • #4
asifadio said:
thanks!
but if i want to convolute with some real function, f(x),
can i just take a real part of
Ak/2 + (Ak/2)e-j2ωt

which is Ak/2 + (Ak/2)(2cos(ωt))?

I hope you mean Ak/2 + (Ak/2)(cos(2ωt)). But what makes you think you can ignore the imaginary part?
 

Related to Trigonometric function and complex exponential

1. What are trigonometric functions?

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratio of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent.

2. What is the unit circle and how is it related to trigonometric functions?

The unit circle is a circle with a radius of 1 centered at the origin of a Cartesian plane. It is related to trigonometric functions because the coordinates of a point on the unit circle can be used to calculate the values of trigonometric functions for that angle.

3. What is a complex exponential?

A complex exponential is a mathematical function of the form e^(ix), where i is the imaginary unit (√-1) and x is a real number. It is a complex number with a real part and an imaginary part that can be represented in polar form as re^(ix), where r is the magnitude and x is the angle.

4. How are trigonometric functions and complex exponentials related?

Trigonometric functions and complex exponentials are related through Euler's formula, which states that e^(ix) = cos(x) + i sin(x). This means that the real and imaginary parts of a complex exponential can be represented as cosine and sine functions.

5. What are some applications of trigonometric functions and complex exponentials?

Trigonometric functions and complex exponentials have many applications in science and engineering, including in the fields of physics, astronomy, and signal processing. They are also crucial in understanding and solving differential equations and in the study of wave phenomena.

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