19.4: Properties of the z-Transform II

The property of Accumulation in signal processing is derived by analyzing the accumulated sum of a discrete-time signal and using the time-shifting property to determine its z-transform. This principle reveals that the z-transform of the summed signal is related to the z-transform of the original signal by a multiplicative factor.

Moreover, the convolution property indicates that the convolution of two signals in the time domain corresponds to the product of their z-transforms in the frequency domain. This property is valid for both causal and noncausal signals. The convolution property can be confirmed by applying the time-shifting property to the corresponding time-domain equation.

The initial value theorem establishes a connection between the initial value of a signal and its z-transform. For a given signal, the initial value can be obtained by evaluating the z-transform as the variable approaches zero. This theorem is particularly useful for determining the starting conditions of a system from its z-transform.

Conversely, the final value theorem determines the final value of a signal by examining its z-transform as the variable approaches one. This theorem is applicable only if the signal continues to exist at infinity and all the poles of the z-transform are within the unit circle, except at the point where the variable equals one.

These properties are crucial for analyzing and designing discrete-time systems. By utilizing the accumulation, convolution, initial value, and final value theorems, the behavior of discrete-time signals, and systems in the z-domain can be studied effectively. Mastery of these properties allows for the manipulation and transformation of signals, aiding in the creation of filters and control systems that function within the discrete-time domain.

The property of Accumulation is derived by expressing the accumulated sum and applying the time-shifting property to solve for the Z-transform.

It states that summing a discrete-time signal produces another signal whose Z-transform equals the Z-transform of the original signal multiplied by z over z minus 1.

The convolution property shows that convolving two signals in the time domain results in the product of their Z-transforms in the frequency domain.

This is valid for both causal and noncausal signals.

Applying the time-shifting property to the time-domain equation helps verify the convolution property.

The initial value theorem relates the initial value of a signal to its Z-transform. For a signal x[n], the initial value is the limit of X(z) as z approaches infinity.

Similarly, the final value theorem states that the final value is the limit of 1 minus the inverse of z multiplied by X(z) as z approaches one.

It applies only if x exists at infinity and all the poles are inside a unit circle except at z  equal to one.