The frequency response of a Discrete-Time system is something experimentally measurable and something that is a complete description of s linear, time-invariant system in the same way that the impulse response is. The frequency response of a linear, time-invariant system is defined as the magnitude and phase of the sinusoidal output of the system with a sinusoidal input. More precisely, if x ( n ) = cos ( ω n ) x ( n ) = cos ( ω n ) and the output of the system is expressed as y ( n ) = M ( ω ) cos ( ω n + φ ( ω ) ) + T ( n ) y ( n ) = M ( ω ) cos ( ω n + φ ( ω ) ) + T ( n ) where T ( n ) T ( n ) contains no components at ω ω , then M ( ω ) M ( ω ) is called the magnitude frequency response and φ ( ω ) φ ( ω ) is called the phase frequency response. If the system is causal, linear, time-invariant, and stable, T ( n ) T ( n ) will approach zero as n → ∞ n → ∞ and the only output will be the pure sinusoid at the same frequency as the input. This is because a sinusoid is a special case of z n z n and, therefore, an eigenvector.

If z z is a complex variable of the special form z = e j ω z = e j ω then using Euler's relation of e j x = cos ( x ) + j sin ( x ) e j x = cos ( x ) + j sin ( x ) , one has x ( n ) = e j ω n = cos ( ω n ) + j sin ( ω n ) x ( n ) = e j ω n = cos ( ω n ) + j sin ( ω n ) and therefore, the sinusoidal input of (3.22) is simply the real part of z n z n for a particular value of z z , and, therefore, the output being sinusoidal is no surprise.