Topic: Communication Systems
The Nyquist filter has impulse response given by
- A. $h(t)=\frac{\sin(\frac{\pi t}{T})}{(\frac{\pi t}{T})}$
- B. $h(t)=\frac{1}{2}(\frac{\sin(\pi t)}{(\pi t)})$
- C. $h(t)=\frac{\sin(\pi t)}{(\frac{\pi t}{T})}$
- D. $h(t)=\frac{\cos(\pi t)}{(\frac{\pi t}{T})}$
Answer: A. $h(t)=\frac{\sin(\frac{\pi t}{T})}{(\frac{\pi t}{T})}$
Explanation: The frequency response of an ideal Nyquist filter is a rectangular pulse: $H(f) = T\text{rect}(fT)$. Using the Fourier Transform pair $\text{sinc}(t) \leftrightarrow \text{rect}(f)$, we get $\text{sinc}(\frac{t}{T}) \leftrightarrow T\text{rect}(fT)$. So, $h(t) = \text{sinc}(\frac{t}{T}) = \frac{\sin(\frac{\pi t}{T})}{(\frac{\pi t}{T})}$.