Question

The impulse response of an LTI system is $h(t)=\delta(t)+0.5\,\delta(t-4)$. If the input is $x(t)=\cos\!\left(\frac{7\pi}{4}t\right)$, the output is_____

• A. $0.5\cos\!\left(\dfrac{7\pi}{4}t\right)$
• B. $1.5\cos\!\left(\dfrac{7\pi}{4}t\right)$
• C. $0.5\sin\!\left(\dfrac{7\pi}{4}t\right)$
• D. $1.5\sin\!\left(\dfrac{7\pi}{4}t\right)$

Hint:

Convolution with $\delta(t-\tau)$ just shifts: $x(t)*\delta(t-\tau)=x(t-\tau)$. For sinusoidal inputs, use the identity $\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$.

Answer 1

The Correct Option is A

Step 1: Use LTI convolution with impulses.
For $h(t)=\delta(t)+0.5\,\delta(t-4)$, \[ y(t)=x(t)*h(t)=x(t)+0.5\,x(t-4). \] Step 2: Substitute $x(t)=\cos(\omega t)$ with $\omega=\dfrac{7\pi{4}$.}
\[ x(t-4)=\cos\!\big(\omega(t-4)\big)=\cos(\omega t-4\omega) =\cos(\omega t)\cos(4\omega)+\sin(\omega t)\sin(4\omega). \] Step 3: Evaluate with $\omega=\dfrac{7\pi{4}$.}
\(4\omega=7\pi\Rightarrow \cos(4\omega)=\cos(7\pi)=-1,\; \sin(4\omega)=\sin(7\pi)=0.\)
Hence \[ y(t)=\cos(\omega t)+0.5\big[-\cos(\omega t)+0 . \sin(\omega t)\big] =\big(1-0.5\big)\cos(\omega t)=0.5\cos(\omega t). \] Final Answer: $0.5\cos\!\left(\dfrac{7\pi}{4}t\right)$