We know that the unit circle must be mapped inside the unit circle.

The map z-1 → g(z-1) must map inside the unit circle in the z-plane into itself to apply digital frequency transformation.

We know that the impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and band pass filters.

Since there is a problem of aliasing in designing high pass and many band pass filters using impulse invariance and mapping of derivatives, we cannot employ the analog frequency transformation followed by conversion of the result into digital domain by use of these two mappings.

As in the analog domain, frequency transformations can be performed on a digital low pass filter to convert it to either a band pass, band stop or high pass filter. The transformation involves the replacing of the variable z-1 by a rational function g(z-1).

We know that the impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and band pass filters due to aliasing problem.

For the map z-1 → g(z-1) to be a valid digital frequency transformation, then the unit circle also must be mapped inside the unit circle.

It is better to perform the mapping from an analog low pass filter into a digital low pass filter by either of these mappings and then perform the frequency transformation in the digital domain because by this kind of frequency transformation, problem of aliasing is avoided.

The value of |ak| < 1 to ensure that a stable filter is transformed into another stable filter to satisfy the condition to satisfy the condition 1.

In the case of bilinear transformation, where aliasing is not a problem, it does not matter whether the frequency transformation is performed in the analog domain or in frequency domain.