Problem 18 - Complex Analysis for Mathematics and Engineering

Question

The Z-transform. Let \(\left\\{a_{n}\right\\}\) be a sequence of complex numbers satisfying the growth condition \(\left|a_{n}\right| \leq M R^{n}\) for \(n=0,1, \ldots\) and for some fixed positive values \(M\) and \(R\). Then the \(Z\) -transform of the sequence \(\left\\{a_{n}\right\\}\) is the function \(F(z)\) defined by \(Z\left(\left\\{a_{n}\right\\}\right)=F(z)=\sum_{n=0}^{\infty} a_{n} z^{-n}\) (a) Prove that \(F(z)\) converges for \(|z|>R\). (b) Find \(Z\left(\left\\{a_{n}\right\\}\right)\) for i. \(a_{n}=2\). ii. \(a_{n}=\frac{1}{n}\). iii. \(a_{\mathrm{n}}=\frac{1}{n+1}\). iv. \(a_{n}=1\), when \(n\) is even, and \(a_{n}=0\) when \(n\) is odd. (c) Prove that \(Z\left(\left\\{a_{n+1}\right\\}\right)=z\left[Z\left(\left\\{a_{n}\right\\}\right)-a_{0}\right] .\) This relation is known as the shifting property for the \(Z\) -transform.

Step-by-Step Solution

Answer

F(z) converges for |z| > R. Solutions: i) \(\frac{2z}{z-1}\), ii-iii) Summation form, iv) \(\frac{z^2}{z^2-1}\). Shifting property is true.

Step 1: Show that \( F(z) \) converges for \( |z|>R \)

To prove this, consider the series \( F(z) = \sum_{n=0}^{\infty} a_n z^{-n} \). We know that \( |a_n| \leq M R^n \). Therefore, \( |a_n z^{-n}| \leq M \left( \frac{R}{|z|} \right)^n \). The geometric series \( \sum_{n=0}^{\infty} \left( \frac{R}{|z|} \right)^n \) converges if \( \left| \frac{R}{|z|} \right| < 1 \), i.e., \( |z| > R \). Thus, by the comparison test, \( F(z) \) converges for \( |z| > R \).

Step 2: Compute \( Z(\{a_n\}) \) for \( a_n=2 \)

For \( a_n = 2 \), the Z-transform becomes \( F(z) = \sum_{n=0}^{\infty} 2 \cdot z^{-n} = 2 \sum_{n=0}^{\infty} z^{-n} \). This is a geometric series with a common ratio of \( z^{-1} \), which converges to \( \frac{2}{1-z^{-1}} = \frac{2z}{z-1} \) for \( |z| > 1 \).

Step 3: Compute \( Z(\{a_n\}) \) for \( a_n=\frac{1}{n} \)

For \( a_n = \frac{1}{n} \), the series is \( \sum_{n=1}^{\infty} \frac{1}{n} z^{-n} \). This series is related to the polylogarithm or the series definition of the logarithm function, and it converges conditionally for \( |z| > 1 \). However, finding an explicit closed form is more involved and not typically expressible in simple terms. It is usually left in summation form.

Step 4: Compute \( Z(\{a_n\}) \) for \( a_n=\frac{1}{n+1} \)

Similarly, for \( a_n = \frac{1}{n+1} \), the transform is \( \sum_{n=0}^{\infty} \frac{1}{n+1} z^{-n} \). This series is another form related to the logarithm series and is not always expressible in elementary functions, but it converges for \( |z| > 1 \) and remains in summation notation.

Step 5: Compute \( Z(\{a_n\}) \) for \( a_n=1 \) when \( n \) is even, \( a_n=0 \) otherwise

Here, the sequence \( a_n = 1 \) when \( n \) is even and 0 when odd. The Z-transform reduces to \( F(z) = \sum_{n=0}^{\infty} z^{-2n} = \sum_{m=0}^{\infty} (z^{-2})^m \). This series converges to \( \frac{1}{1-z^{-2}} = \frac{z^2}{z^2-1} \) for \( |z| > 1 \).

Step 6: Prove the Shifting Property

To show that \( Z(\{a_{n+1}\}) = z[Z(\{a_n\}) - a_0] \), consider the new sequence \( \{a_{n+1}\} \). Its Z-transform is \( \sum_{n=0}^{\infty} a_{n+1} z^{-n} \). Shift the index by 1: it becomes \( \sum_{n=1}^{\infty} a_{n} z^{-(n-1)} = z \sum_{n=1}^{\infty} a_n z^{-n} = z(Z(\{a_n\}) - a_0) \). Thus, the property is proved.

Key Concepts

Complex Sequences

In mathematics, a sequence of complex numbers \(\{a_n\}\) involves elements that have both real and imaginary parts. This allows for increased flexibility in many types of calculations, especially in fields such as engineering and physics.
Complex sequences can be expressed using the format \(a_n = x_n + iy_n\), where \(x_n\) and \(y_n\) are real numbers, and \(i\) represents the imaginary unit, satisfying \(i^2 = -1\).
When dealing with complex sequences, we apply the growth condition. This condition ensures that the absolute value, \(|a_n|\), is bounded by a constant multiplied by a power of a fixed base.
For example, if a sequence satisfies \(|a_n| \leq M R^n\), it implies that each term isn't growing faster than the rate determined by \(R^n\). By evaluating Z-transforms on such sequences, we can unlock a deeper understanding of signal processing through translating sequences into the complex frequency domain.

Convergence Analysis

Convergence analysis in the context of the Z-transform is crucial for understanding where the infinite series actually sums up to a finite value. When we talk about the convergence of a Z-transform, we refer to a situation where the series \(F(z) = \sum_{n=0}^{\infty} a_n z^{-n}\) sums to limit within a particular region in the complex plane.
The condition \(|z| > R\) indicates that the series converges outside a circle with radius \(R\) in the complex plane, known as the region of convergence (ROC).
To determine convergence:

• Compare the series terms to a known convergent series (e.g., a geometric series).
• Verify that the difference between individual terms diminishes as \(n\) increases.

By establishing convergence, we can apply various mathematical tools and transform a complex problem into a more manageable one.

Shifting Property

The shifting property of the Z-transform is a valuable tool in signal processing. It is described by the relation \(Z(\{a_{n+1}\}) = z[Z(\{a_n\}) - a_0]\). This property helps us understand how the Z-transform of a sequence reacts to changes in the sequence's index.
Essentially, if you "shift" the index of the sequence by 1, the Z-transform formula adapts accordingly by including a factor of \(z\) and adjusting by \(a_0\), the initial element of the sequence.
Here's how to apply it:

• Begin with the original sequence \(\{a_n\}\) and its Z-transform.
• If the sequence is shifted (e.g., \(a_n ightarrow a_{n+1}\)), incorporate the shifting property into your calculations.
• This process can simplify recursive sequences or systems with delays, such as digital filters.

The shifting property provides a framework for handling changes in input sequences in a streamlined manner, maintaining the integrity and efficiency of signal analysis.