i. Time Shifting

ii. Differentiation in Time Domain

Mumbai University > Information Technology > Sem 3 > Principles of analog and digital communication

Marks:- 5M

Year:- Dec 2015

1. Time shifting:

• The time shifting property states that if x(t) and X(f) form a Fourier transform pair then,

x (t-$t_d$) F ↔$e^{-j2πf{t_d}} $ X (f)

Here the signal x(t- $t_d$ ) is a time shifted signal. It is the same signal x(t) only shifted in time.

• Proof:

F[x (t-$t_d$)] = $∫_{-∞}^∞ x (t-{t_d}) {e^{-j2πft}}$ dt…………………………………….(1)

Let (t-$t_d$) = τ ,

∴ t = $t_d$+ τ

∴dt = dτ

Substituting these values in equation (1) we get,

F[x (t-$t_d$)] =$∫_{-∞}^∞ x (τ).$$e^{-j2πf({t_d}+τ)} dτ$ = $e^{-j2πf{t_d}} {∫_{-∞}^∞ x (τ)} {e^{-j2πft}}dτ$ ∴ F[x (t-$t_d$)] = $e^{-j2πf{t_d}}$ X (f)…………………………………………Proved 2. **Differentiation in Time Domain:** - Some processing techniques involve differentiation and integration of the signal x(t).This property is applicable if and only if the derivative of x(t) is Fourier transformable. - **Statement:** Let x(t) F↔ X(f) and let the derivative of x(t) be Fourier transformable. Then, d/dtx(t) F ↔j2πfX(f) **Proof:** By the definition of inverse Fourier transform, x (t) = $∫_{-∞}^∞ X (f) {e^{j2πft}}$ df ∴d/dtx(t) =d/dt [$∫_{-∞}^∞ X(f) {e^{j2πft}}df$ ] = $∫_{-∞}^∞ X(f) (d/( dt) {e^{j2πft}}$ df d/dtx(t) =$∫_{-∞}^∞ [X(f) .j2πf] {e^{j2πft}}df$

• As per the definition of the inverse Fourier transform the term inside the square bracket must be the Fourier transform of d/dt x(t)

∴ F[d/dtx(t)] =j2πfX(f)

Or d/dtx(t) F j2πfX(f)…………………………………………….Proved