The Parks-McClellan algorithm is a widely used algorithm for designing optimal FIR filters. It is commonly used to design filters with specific frequency response characteristics, such as magnitude response, phase response, or both.

In the case of a Hilbert transformer, the desired frequency response is a phase shift of -90 degrees for all frequencies. This means that the filter should introduce a phase shift of -90 degrees to the input signal at all frequencies.

To achieve this, the Parks-McClellan algorithm uses a concept known as the "deviation" or "weight" function. This function specifies the desired frequency response at different frequencies. In the case of a Hilbert transformer, the deviation function will be a constant -90 degrees for all frequencies.

The Parks-McClellan algorithm also uses a parameter called "JTYPE" to specify the type of filter to be designed. The JTYPE parameter controls the shape of the frequency response and can take different values for different types of filters.

For a Hilbert transformer, the value of JTYPE is 3. This value corresponds to a type III linear phase filter, which is suitable for designing a Hilbert transformer. The type III linear phase filter has a constant group delay, which means that it introduces a constant phase shift to all frequencies.

The Parks-McClellan algorithm uses an iterative optimization procedure to design the filter. It starts with an initial guess for the filter coefficients and iteratively refines them to minimize the error between the desired frequency response and the actual frequency response of the filter. The algorithm takes into account the specified deviation function and the JTYPE parameter to guide the optimization process.

In summary, for a Hilbert transformer designed using the Parks-McClellan algorithm, the value of JTYPE is 3. This value specifies the type of filter to be designed and helps in achieving the desired phase response of -90 degrees for all frequencies.