Here is a detailed solution. If you like the response, a tip is appreciated.
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Z-Transform
The Z-transform of the signal S(n) = [-1, 0, 2, -1] is given by:
\[ S(z) = -1z^3 + 2z^2 - z + 1.\]
As you have already figured out. This is obtained by taking the sum of each sample of the signal multiplied by z to the power of its corresponding sample index, which is given by the formula:
\[ S(z) = \sum_{n=0}^{\infty} S(n) z^{-n}\]
where $S(n)$ is the sample at time n and z is the complex variable used in the Z-transform.
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Amplitude-Frequency Characteristic:
The amplitude-frequency characteristic of a signal is a plot of the magnitude of the frequency response of the signal. In the case of a discrete-time signal like $S(n) = [-1, 0, 2, -1]$, the amplitude-frequency characteristic can be obtained using the discrete-time Fourier transform (DTFT), which provides the frequency response of a signal in the continuous frequency domain.
In other words, the amplitude-frequency characteristic of a signal gives the magnitude of its Z-transform X(z) at different frequencies. The magnitude of $X(z)$ is given by:
\[ |X(z)| = \sqrt{ real(X(z))^2 + imag(X(z))^2}\]
So, to calculate the amplitude-frequency characteristic of the signal $S(n) = [-1, 0, 2, -1]$, you would first need to obtain its frequency response.
This can be done by computing the DTFT of the signal using the formula:
\[X(\omega) = \sum_{n=0}^{\infty} S(n)e^{-j \omega n}\]
where $X(\omega)$ is the DTFT, $\omega$ is the continuous frequency variable, and $S(n)$ is the sample at time n.
Once the frequency response is obtained, you can compute the magnitude of $X(\omega)$ to obtain the amplitude-frequency characteristic of the signal, which is simply the magnitude spectrum of the signal.
The magnitude spectrum is given by:
\[ |X(\omega)| = \sqrt{Re^2[X(\omega)] + Im^2[X(\omega)]}\]
where $Re[X(\omega)]$ and $Im[X(\omega)]$ are the real and imaginary parts of $X(\omega)$, respectively. The magnitude of these coefficients represents the amplitude of the frequency response of the signal at each of the frequencies.
Finally, you can plot the magnitude spectrum as a function of frequency, typically on a logarithmic scale, to obtain the amplitude-frequency characteristic of the signal.
This can be done using various software tools, such as MATLAB
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Amplitude-Phase Characteristic
Similar to the above, the amplitude-phase characteristic can be obtained using the discrete Fourier transform (DFT).
The DFT of a signal x(n) of length N is given by:
\[ X[k] = \sum_{n=0}^{N-1} x(n)e^{-j 2 \pi k n/N}\]
where $X[k]$ is the kth DFT coefficient, $j$ is the imaginary unit, and $k$ and $n$ are indices ranging from $0$ to $N-1$.
The magnitude of the DFT coefficients can be computed as:
\[ |X[k]| = \sqrt{(Re{X[k]})^2 + (Im{X[k]})^2}\]
where Re{X[k]} and Im{X[k]} are the real and imaginary parts of X[k], respectively. The magnitude of the DFT coefficients represents the amplitude of the frequency response of the signal at each of the frequencies k/N.
The phase of the DFT coefficients can be computed as:
\[\phi[k] = atan2(Im{X[k]}, Re{X[k]})\]
where atan2 is the two-argument inverse tangent function. The phase of the DFT coefficients represents the phase of the frequency response of the signal at each of the frequencies k/N.
The amplitude-phase characteristic of the signal can be obtained by plotting the magnitude and phase of the DFT coefficients as a function of frequency.
The offered bounty is too low for the level of the question.
Very low bounty!
I can provide you guidance and details of how to obtain the transform and the two characteristics, but cannot draw the plots. Let me know if that works. If you want the plots, they need to be plotted using the magnitude of the phase at different frequencies.
That works for me