Find amplitude-frequency characteristic of a discrete finite signal using Z-transform

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.