CTFT of the convolution of some signals

Find the CTFT of the convolution of 10sin(t)with 2δ(t + 4).

Question

CTFT of the convolution of some signals

Find the CTFT of the convolution of 10sin(t)with 2δ(t + 4).

Accepted Answer

Method 1: Do the convolution first and find the CTFT of the result.

[latex]10sin(t)^∗ 2δ(t + 4)[/latex] = 20sin(t + 4)

Apply the time-shifting property.

[latex]20sin(t + 4) \overset{\mathcal{F}}{\longleftrightarrow } j20π [δ (ω + 1) − δ( ω− 1)]e^{j4ω}[/latex]

or

[latex]20sin(t + 4)\overset{\mathcal{F}}{\longleftrightarrow } j10[δ ( f + 1/2π ) − δ( f − 1/2π )]e^{j8π f}[/latex]

Method 2: Do the CTFT first to avoid the convolution.

[latex]10sin(t)^∗ 2δ(t + 4)\overset{\mathcal{F}}{\longleftrightarrow }\mathcal{F}(10sin(t))\mathcal{F}(2δ(t + 4)) = 2{\mathcal{F}}(10sin(t))\mathcal{F}(δ(t))e^{j4ω}[/latex]

[latex]10sin(t)^∗ 2δ(t + 4)\overset{\mathcal{F}}{\longleftrightarrow }j20π[δ(ω+1)-δ(ω-1)e^{j4ω}[/latex]

or

[latex]10sin(t)^∗ 2δ(t + 4)\overset{\mathcal{F}}{\longleftrightarrow }\mathcal{F}(10sin(t)){\mathcal{F}}(2δ(t + 4)) = 2\mathcal{F}(10sin(t))\mathcal{F}(δ(t))e^{j8πf}[/latex]

[latex]10sin(t) ^∗ 2δ(t + 4)\overset{\mathcal{F}}{\longleftrightarrow }j10[δ ( f + 1/2π ) − δ( f − 1/2π )]e^{j8π f}[/latex].

Question 6.14: CTFT of the convolution of some signals Find the CTFT of the......

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