Unit 6: The Z-transform - Subjective Questions

The z-transform converts a discrete-time signal into a complex frequency-domain representation.

Bilateral (two-sided) z-transform:

Unilateral (one-sided) z-transform:

where is a complex variable.

Key points:

• The z-transform is the discrete-time counterpart of the Laplace transform.
• When (i.e. ), it reduces to the Discrete-Time Fourier Transform (DTFT).
• The unilateral form is useful for solving difference equations with initial conditions.
• The region of convergence (ROC) must always accompany for a complete description.

The DTFT is a special case of the z-transform evaluated on the unit circle.

z-transform:

Substituting :

This is the DTFT of the sequence .

Special case (, i.e. ):

Key observations:

• The DTFT exists only if the unit circle lies within the ROC.
• The factor acts as a convergence-forcing term, allowing the z-transform to exist for signals whose DTFT does not converge.
• The z-plane's unit circle corresponds to the frequency axis of the DTFT.

The Region of Convergence (ROC) is the set of values of in the complex plane for which the z-transform sum converges (i.e. is finite):

Properties of the ROC:

• The ROC is an annular region (ring) centered at the origin: .
• The ROC does not contain any poles.
• For a finite-duration sequence, the ROC is the entire z-plane, possibly excluding and/or .
• For a right-sided sequence, the ROC is the region outside the outermost pole: .
• For a left-sided sequence, the ROC is the region inside the innermost pole: .
• For a two-sided sequence, the ROC is a ring between two poles.
• The DTFT converges if and only if the ROC includes the unit circle.
• The ROC is a connected region.

Given (a right-sided/causal signal).

Applying the definition:

This is a geometric series that converges when , i.e. .

Summing the geometric series:

Result:

• z-transform:
• Pole: at
• Zero: at
• ROC: (outside the pole, as expected for a causal signal)

If , the ROC includes the unit circle, so the DTFT exists.

Both sequences yield the same algebraic expression for the z-transform but have different ROCs, illustrating why the ROC is essential.

For (right-sided):

For (left-sided):

Comparison table:

Conclusion: Since both share the same , the ROC uniquely determines the time-domain signal. The z-transform expression alone is insufficient.

1. Linearity Property:

If and , then:

Proof:

The ROC is at least the intersection of the two individual ROCs.

2. Time-Shifting Property:

If , then:

Proof:

Let , so :

The ROC is unchanged (except possibly at or ).

Convolution Property:

If and , then:

Derivation:
The convolution is defined as:

Taking the z-transform:

Interchanging the order of summation:

Using the time-shift property, :

Significance:

• Convolution in time domain becomes multiplication in the z-domain, greatly simplifying analysis.
• The system transfer function is .
• The ROC of contains at least the intersection of the ROCs of and .

(i) Scaling in the z-domain (Multiplication by exponential):

Multiplying by in time scales the z-plane; poles and zeros are scaled by . The ROC is scaled: if original ROC is , new ROC is .

(ii) Time reversal:

A right-sided signal becomes left-sided. If original ROC is , new ROC is .

(iii) Differentiation in the z-domain:

Proof of (iii): Starting from , differentiate:

Rearranging gives . The ROC remains the same.

These theorems apply to causal sequences (right-sided, for ).

Initial Value Theorem:

Proof:

As , all terms with () vanish, leaving .

Final Value Theorem:

Condition: All poles of must lie inside the unit circle (i.e. the limit must exist / system stable).

Proof (sketch): Using the z-transform of :

Taking and evaluating the telescoping sum gives , from which the result follows.

Use: These provide steady-state and starting values without computing the full inverse transform.

The inverse z-transform recovers from and its ROC. The main methods are:

1. Partial Fraction Expansion (PFE):

• Express as a sum of simple first/second-order terms.
• Use standard transform pairs (with ROC) to invert each term.
• Most widely used for rational .

2. Power Series Expansion (Long Division):

• Expand as a power series in (causal) or (anti-causal).
• The coefficients directly give .
• Useful when a closed form is not needed.

3. Contour Integration (Residue Method):

• Based on the Cauchy residue theorem.
• Mathematically rigorous but computationally involved.

4. Look-up Table Method:

• Manipulate using properties into recognizable standard forms and use known pairs.

Note: The ROC is essential in all methods to obtain the correct (unique) time-domain sequence.

Step 1: Partial Fraction Expansion

Solving for A (multiply by and set ):

Solving for B (set ):

Step 2: Write the expansion

Step 3: Invert each term

Since ROC is (outside all poles), both terms are causal:

Final Answer:

The transfer function of an LTI system is the z-transform of its impulse response , and equals the ratio of output to input transforms:

Obtaining from a difference equation:

Consider a general LCCDE:

Taking the z-transform (using the time-shift property ):

Solving for the ratio:

Key points:

• is a rational function — a ratio of polynomials in .
• The roots of the numerator are the zeros; the roots of the denominator are the poles.
• Poles, zeros, and the ROC completely characterize the system behaviour (stability, causality).

Causality Condition:

• An LTI system is causal if its impulse response satisfies for (right-sided).
• In the z-domain, the ROC is the region outside the outermost pole, i.e. , and includes .
• For a rational , causality requires the order of the numerator order of the denominator (in ).

Stability Condition (BIBO):

• A system is BIBO stable if .
• This is equivalent to the ROC including the unit circle .

Combined Causality + Stability:

• For a system to be both causal and stable, the ROC must be and include the unit circle.
• This requires all poles to lie inside the unit circle ().

Summary Table:

Step 1: Transfer Function

Taking the z-transform of the difference equation:

Step 2: Find Poles (factor denominator)

Poles at and .

Step 3: Partial Fractions

Step 4: Impulse Response (causal, ROC )

Step 5: Stability
Both poles ( and ) lie inside the unit circle, so the system is BIBO stable.

A pole-zero plot displays the locations of poles () and zeros () of in the complex z-plane, along with the unit circle for reference.

Significance:

• Provides a complete graphical characterization of the system (up to a gain constant).
• Reveals stability, causality, and frequency-response behaviour at a glance.

Relation to Stability:

• For a causal system, if all poles lie inside the unit circle, the system is stable.
• Poles on the unit circle → marginally stable; poles outside → unstable.

Relation to Frequency Response:

• The frequency response is , obtained by evaluating on the unit circle.
• Poles near the unit circle cause peaks (resonances) in the magnitude response at the corresponding frequency.
• Zeros near/on the unit circle cause dips (nulls) in the magnitude response.
• Geometrically:
  

Thus the pole-zero geometry directly shapes filter behaviour (low-pass, high-pass, band-pass, etc.).

Similarities:

• Both are generalizations of Fourier analysis providing a complex-frequency domain representation.
• Both convert convolution into multiplication, simplifying LTI system analysis.
• Both use a transfer function with poles and zeros to characterize systems.
• Both require a region of convergence for a complete/unique definition.

Differences:

Mapping: The transformation maps the s-plane to the z-plane — the imaginary axis maps to the unit circle and the left-half plane maps to the interior of the unit circle.

Step 1: Use Euler's identity

So:

Step 2: Use the pair

With and (both with , ROC ):

Step 3: Combine over a common denominator

Numerator: .
Denominator: .

Final Result:

Software tools like MATLAB, Octave, or Python (SciPy) allow numerical representation and visualization of discrete-time systems.

System Representation:

• A system is represented by numerator and denominator coefficient vectors, e.g. b = [b0, b1, ...] and a = [a0, a1, ...].

Key MATLAB functions:

• tf(b, a, Ts) — creates a discrete transfer function object.
• zplane(b, a) — plots the pole-zero diagram with the unit circle.
• roots(a) / roots(b) — computes poles / zeros directly.
• freqz(b, a) — computes and plots the frequency response .
• impz(b, a) — computes the impulse response.
• residuez(b, a) — performs partial fraction expansion.

Python (SciPy) equivalents:

• scipy.signal.tf2zpk(b, a) — returns zeros, poles, gain.
• scipy.signal.freqz, scipy.signal.dimpulse.

Steps for Pole-Zero Analysis:

1. Enter coefficient vectors b and a.
2. Compute poles and zeros using roots/tf2zpk.
3. Plot with zplane to visualize positions relative to the unit circle.
4. Interpret: poles inside unit circle → stable; poles near circle → resonant peaks.
5. Verify with freqz and impz plots.

This lets engineers quickly assess stability, causality, and filter behaviour without lengthy hand calculations.

The following are frequently used standard z-transform pairs:

Note: For each pair the ROC must be stated since different signals can share the same algebraic expression. These pairs, combined with z-transform properties, allow inversion of most rational transforms.

Step 1: Factor the denominator

Poles at and .

Step 2: Partial fractions


So .

Case 1: ROC (both terms right-sided / causal)

This is a causal but not stable signal (pole at on the unit circle).

Case 2: ROC (two-sided)

• Pole at term becomes left-sided; pole at stays right-sided:
  

Discussion:

• The same produces different sequences depending on ROC.
• Only the ROC determines whether each pole contributes a right- or left-sided term.
• Neither case includes the unit circle in a stable manner, so no stable solution exists here.