# What does a shift in time domain do to the phase in frequency domain?

Suppose $y_k = x_{(k-m)}$, i.e. $x$ is a time shifted version of $y$. Then the Z transform of $y$ is $z^{-m}X(z)$.

What does multiplying by $z^{-m}$ mean in terms of the phase response of $X$ for a negative shift? A positive shift?

$z^{-1}$ is a delay of $T$, where $T$ is the sampling increment. This is equivalent to the Laplace operator, $e^{-sT}$, which transforms to the frequency domain as $e^{-jwT}$. The phase contributed by this function at the frequency $w$ is $-wT$ radians, as $e^{-jwT} = \cos(wT)-j\sin(wT)$. So the phase contributon of $z^{-m}$ is $-mwT$ radians. The gain is not affected since the magnitude of $e^{-jmwT}$ is unity.