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Let the discrete-time signal be defined by z[n], where z[n] is specified by given values, and the remaining values are zero. The task involves expressing z[n] as a sum of step functions, analyzing whether z[n] is an energy or a power signal, and computing its corresponding energy or power if applicable. Additionally, the causal nature of the reversed and shifted signal z[−n + 7] must be examined.

In the second part, the given discrete-time signal z[n] is depicted graphically. The objectives include expressing the signal as a sum of unit step functions, sketching a new signal y[n] which is a weighted sum involving shifts and sums of z[n], and analyzing the signal's energy or power characteristics.

Paper For Above instruction

The analysis of discrete-time signals such as z[n] involves understanding their structure through mathematical representations and energy/power content. This exploration begins by expressing the signal as a sum of step functions, followed by an assessment of its energy or power classification, and concludes with examining causality when the signal is reversed and shifted.

Firstly, expressing z[n] as a sum of step functions u[n] requires understanding the nature of the defined values of z[n]. Suppose z[n] is non-zero over a finite interval; in that case, it can be represented as a linear combination of shifted unit step functions u[n - n₀], where n₀ marks points where z[n] changes value. For example, if z[n] is defined as a constant c over the interval n₁ ≤ n ≤ n₂ and zero elsewhere, then z[n] can be written as:

z[n] = c [u[n - n₁] - u[n - n₂ - 1]]

This representation captures the onset and offset of the signal in terms of step functions. More complex signals with multiple changes can be expressed as sums of such step functions, each corresponding to a change point in the signal values.

Secondly, the classification of z[n] as an energy or power signal depends on the calculation of its energy and power. The energy E of a discrete-time signal z[n] is given by:

E = Σ_{n=-∞}^{∞} |z[n]|^2

If this sum is finite, then z[n] is an energy signal; if infinite, it is not. The average power P is given by:

P = lim_{N→∞} (1/2N+1) Σ_{n=-N}^{N} |z[n]|^2

If P is finite and non-zero, z[n] is a power signal. Otherwise, if both energy and power are zero, or the energy sum diverges, the signal is either neither or requires further analysis.

Thirdly, the causality aspect of the signal z[−n + 7] relates to whether the signal depends only on present and past values (causal) or includes future values (non-causal). A signal z[n] is causal if z[n] = 0 for all n

z[−n + 7] = z[−(n - 7)]

which is causal if it equals zero for all n 7, the original z[n] must be zero for n > 7, so causality depends on the support of z[n].

In the second part, the specific signal z[n] is provided as a graphical plot. To express it as a sum of step functions, segments where the signal changes value are identified, and each change is represented by a scaled and shifted u[n] function. For example, a block that is active over a certain interval can be represented as a difference of step functions, effectively turning the pulsed or piecewise constant signal into a sum of step functions.

The construction of y[n] involves summing weighted shifts of z[n], indicated by the sum y[n] = Σ_{m = -1}^{1} (m + 1) z[n - 3m], which entails shifting z[n] by multiples of 3 and applying weights (−1 + 1, 0, 1 + 1). The resulting y[n] can be sketched by computing each component and superimposing these shifted signals, with axes labeled appropriately.

Finally, an analysis of the energy or power of z[n], similar to the first part, determines whether the signal is finite in energy or power. If the sum of squared magnitudes over time converges, z[n] qualifies as an energy signal; if the average power converges to a finite constant, it is a power signal. Should neither condition hold, it indicates that the signal is neither, such as in the case of persistent non-zero energy over an infinite interval without finite average power.

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