i. Time Shifting

ii. Differentiation in Time Domain

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 ↔$e^{-j2πf{t_d}} X (f)** Here the signal x(t- td ) is a time shifted signal. It is the same signal x(t) only shifted in time. **Proof:** F[x (t-$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

1. 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 x(t)/dt F ↔j2πfX(f)

Proof:

By the definition of inverse Fourier transform,

x (t) = $∫_{-∞}^∞ X (f) {e^{j2πft}}df$

∴d/dtx(t) =d$[∫_{-∞}^∞ X(f) {e^{j2πft}}df ]/dt$

= $∫_{-∞}^∞ X(f)(d/(dt) {e^{j2πft}} df$

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 dx(t)/dt

∴ F[d/dtx(t)] =j2πfX(f)

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