Question

The signal x[n] = sin(ππ/6)/(ππ) is processed through a linear filter with the impulse response h[n] = sin(ω_cn) /(π) where ω_c > π/6. The output of the filter is

1. sin (2ω_cn)/(ππ)
2. sin (ππ/3)/(ππ)
3. [sin(ππ/6)/(ππ)]²
4. sin(ππ/6)/(ππ)

Answer 1

Correct Answer - Option 4 : sin(ππ/6)/(ππ)

\(x\left[ n \right] = \frac{{\sin \left( {\frac{{n\pi }}{6}} \right)}}{{n\pi }}\)

\(h\left[ n \right] = \frac{{\sin \left( {{\omega _c}n} \right)}}{\pi },{\omega _c} > \frac{\pi }{6}\)

output, y[n] = x[n] * h[n]

in frequency domain both will get multiplied

\(y\left[ n \right] = \frac{{\sin \left( {\frac{{n\pi }}{6}} \right)}}{{n\pi }}.\frac{{\sin \left( {{\omega _c}n} \right)}}{\pi }\)

\(= \frac{{\sin \left( {\frac{{n\pi }}{6}} \right)}}{{n\pi }}\;as\;{\omega _c} > \frac{\pi }{6}\)