Given a linear time-invariant system, prove the properties below:

(a) A constant group delay is a necessary but not sufficient condition for the delay introduced by the system to a sinusoid to be independent of its frequency.

(b) Let \(y_{1}(n)\) and \(y_{2}(n)\) be the outputs of the system to two sinusoids \(x_{1}(n)\) and \(x_{2}(n)\) respectively. A constant group delay \(\tau\) implies that if \(x_{1}\left(n_{0}\right)=x_{2}\left(n_{0}\right)\), then \(y_{1}\left(n_{0}-\tau\right)=y_{2}\left(n_{0}-\tau\right)\).