Subject : Signals & Systems

Topic : Continuous Time Fourier Transform (CTFT) and Discrete Time Fourier Transform (DTFT).

Difficulty: Medium

Time scaling:

Compression of a signal in time domain is equivalent to expansion in frequency domain and vice versa. Proof:

Proof: $Y(ω)=\int_{-∞}^∞ y(t) e^{-jωt} \, dτ$

$Y(ω)=\int_{-∞}^∞ x(at) e^{-jωt} \, dτ$

Put at = τ then t = τa ∴dt = 1a dτ and limits will remain same

$Y(ω)=\int_{-∞}^∞ x(τ) e^{-jω τ/a} (1/a) \, dτ$

$=1a \int_{-∞}^∞ x(τ) e^{-j ω/a τ}\, dτ$

∴Y(ω) = 1a X(ω/a)

Amplitude scaling:

Amplitude scaling is a very basic operation performed on signals to vary its strength. It can be mathematically represented as y(t) = a x(t).

Here a is the scaling factor where

$a \lt 1 \quad\longrightarrow\quad \ signal\ is \ attenuated$

$a \gt 1 \quad\longrightarrow\quad \ signal\ is \ amplified$

$= \int_{-∞}^∞ a x(t) e^{-jωt} \, dτ$

$=a \int_{-∞}^∞ x(t) e^{-jωt} \, dτ$

∴Y(ω) = a X(ω)