Answer:

The given system function H(z) = z - k represents an all-pass filter. Let's understand why this is the correct answer.

1. All-Pass Filter:
An all-pass filter is a type of filter that allows all frequencies to pass through without attenuation but alters the phase response. In other words, the amplitude of the input signal remains unchanged, but the phase of certain frequency components is modified.

2. System Function:
The system function of a filter is a mathematical representation that relates the input signal to the output signal. It is expressed in terms of the complex variable z, which represents the unit delay operator.

In this case, the given system function H(z) = z - k represents a simple first-order difference equation. The input signal is multiplied by 'z' (unit delay), and 'k' is subtracted from it. This means that the output signal is the input signal delayed by one sample and then subtracted by a constant 'k'.

3. Frequency Response:
To analyze the frequency response of the given system function, we substitute z = e^(jω), where ω is the angular frequency.

H(z) = z - k
H(e^(jω)) = e^(jω) - k

The frequency response H(e^(jω)) can be represented as a complex number with magnitude and phase:

|H(e^(jω))| = sqrt( (Re{H(e^(jω))})^2 + (Im{H(e^(jω))})^2 )
arg{H(e^(jω))} = atan2(Im{H(e^(jω))}, Re{H(e^(jω))})

4. Magnitude and Phase Response:
For the given system function H(e^(jω)) = e^(jω) - k, the magnitude and phase responses are as follows:

|H(e^(jω))| = sqrt( (cos(ω) - k)^2 + sin^2(ω) )
arg{H(e^(jω))} = atan2(sin(ω), cos(ω) - k)

From the magnitude response, we can observe that the amplitude of the input signal remains unaffected, which is a characteristic of an all-pass filter. The phase response shows that the phase shift depends on the value of 'k'. Different values of 'k' will result in different phase shifts.

Conclusion:
Based on the analysis of the system function and frequency response, we can conclude that the given filter with the system function H(z) = z - k is an all-pass filter. It does not attenuate any frequencies but alters the phase response of the input signal.