Question

Consider an interval Δt of an FM wave s(t) = Acosθ(t) such that θ(t) satisfies the condition θ(t + Δt) − θ(t) = π. Show that if Δt is small, the instantaneous frequency fᵢ of the FM wave inside this interval is approximately fᵢ ≈ 1/2Δt. Hint: consider the mathematical definition of differentiation.

Answer 1

The instantaneous frequency fi of an FM wave for a small interval ∆t, where a phase change of π occurs, can be approximated using differentiation, with fi being approximately equal to 1/2∆t.

Step-by-step explanation:

To demonstrate the relationship between the interval ∆t and the instantaneous frequency fᵢ of the FM wave s(t) = Acosθ(t), we start by considering the given condition θ(t + ∆t) − θ(t) = π. If ∆t is small enough, we can make an approximation using the concept of differentiation. The phase change over the interval is equivalent to the derivative of θ(t) multiplied by ∆t, which can be expressed as dθ/dt • ∆t ≈ π

The instantaneous frequency is defined as fᵢ = 1/(2π) • dθ/dt, substituting the approximate value from the phase change we get fᵢ ≈ 1/(2π) • (π/∆t) = 1/2∆t. Hence, for a small interval ∆t, the instantaneous frequency of an FM wave can be approximately expressed as fᵢ ≈ 1/2∆t.