Question 6.12: CTFT of some time-scaled and time-shifted sines If x(t) = 10......

(a) In this example the cyclic frequency of the sinusoid is 1/2π and the radian frequency is 1.
Therefore the numbers will be simpler if we use the radian-frequency form of the CTFT.
Using the linearity property and looking up the transform of the general sine form,

(b) From part (a), 10sin(t)\xleftrightarrow{\mathcal{F}}j10π[δ(ω+1)-δ(ω-1)]. Using the time scaling property,

10sin(2t) \xleftrightarrow{\mathcal{F}} j5π[δ(ω/2+1)-δ(ω/2-1)].

Then, using the time-shifting property,

10sin(2(t − 1))\xleftrightarrow{\mathcal{F}}j5π[δ(ω/2+1)-δ(ω/2-1)]e^{-jω}.

Then, using the scaling property of the impulse,

or

(c) From part (a), 10sin(t)\xleftrightarrow{\mathcal{F}}j10π[δ(ω+1)-δ(ω-1)].

Applying the time-shifting property first

10sin(t-1)\xleftrightarrow{\mathcal{F}}j10π[δ(ω+1)-δ(ω-1)]e^{-jω}.

Then, applying the time-scaling property,

Then, using the scaling property of the impulse,

or