Signal strength of a wave packet

[Elvis 123456789]

Homework Statement

Assume a wave packet is has contributions from various frequencies, give by g(ω)=C for |ω|<ω0, and g(ω) =0 for elsewhere.

a)What is the signal strength as a function of time, i.e., V(t)=?

b) Sketch g(ω) and V(t); You can use fooplots.com, for example, or python.

c) Indicate Δω and Δt in the above plots; Does the products of these two satisfy ΔωΔt>1/2?

Homework Equations

V(t) = 1/√(2π) * integral from -∞ to +∞ of [ g(ω)*exp(iωt)] dω

exp(iωt) = cos(ωt) + isin(ωt)

The Attempt at a Solution

a.) V(t) = 1/√(2π) * integral from -ω₀ to +ω₀ of [ C*exp(iωt)] dω

using eulers formula and the properties of even and odd functions

V(t) = 2/√(2π) * integral from 0 to +ω₀ of [ C*cos(ωt)] dω

V(t) = 2C/√(2π) * sin(ω₀t)/t = √(2/π)*ω₀C * sin(ω₀t)/ω₀t

b.) the sketches for g(ω) and V(t) are in the attachments

c.) I am not really sure what Δω and Δt in these graphs

 

[blue_leaf77]

V(t) = 2/√(2π) * integral from 0 to +ω0 of [ C*cos(ωt)] dω

Shouldn't the lower integral limit be $-\omega_0$?

c.) I am not really sure what Δω and Δt in these graphs

Use the definition of standard deviation. For example for V(t), you will use
$$
\Delta t = \sqrt{E[t^2]-(E[t])^2}
$$
where $E[\,\,]$ means taking average over the intensity $|V(t)|^2$. Similar arguments for $g(\omega)$.

 

[Elvis 123456789]

Shouldn't the lower integral limit be −ω0−ω0-\omega_0?

I have a factor of 2 in the front to account for that. Is that not right?

 

[blue_leaf77]

I have a factor of 2 in the front to account for that. Is that not right?

Ah sorry I missed that, you are right.