Aliasing occurs if the sampling rate Fs is less than twice the highest frequency contained in X(F).

The digital filter with frequency response H(ω) has the frequency response characteristics of the corresponding analog filter if the sampling interval T is selected sufficiently small to completely avoid or at least minimize the effects of aliasing.

In the given diagram, the continuous line is the frequency response of analog filter and dotted line is the frequency response of the corresponding digital filter with aliasing.

It is clear that the impulse invariance method is in -appropriate for designing high pass filter due to the spectrum aliasing that results from the sampling process.

We know that z=esT

In the impulse invariance method, our objective is to design an IIR filter having a unit sample response h(n) that is the sampled version of the impulse response of the analog filter. That is

The above given frequency response depicts the frequency response of a low pass analog filter and the frequency response of the corresponding digital filter.

When a continuous time signal x(t) with spectrum X(F) is sampled at a rate Fs=1/T samples per second, the spectrum of the sampled signal is periodic repetition.

When a continuous time signal x(t) with spectrum X(F) is sampled at a rate Fs=1/T samples per second, the spectrum of the sampled signal is periodic repetition of the scaled spectrum Fs.X(F).

In the impulse invariance method, the normalized frequency f is given by