3. Discrete-Time LTI Systems

Introduction and Context

A discrete-time signal-processing system transforms a sequence of input samples into an output sequence. In digital signal processing, the most useful class of systems is the linear time-invariant (LTI) system. LTI systems are important because they combine mathematical tractability with broad engineering relevance. Digital filters, communication receivers, control algorithms, audio processors, biomedical monitors, and image-processing operations are commonly modeled or implemented as discrete-time LTI systems.

The defining properties are linearity and time invariance. Linearity permits the decomposition of complicated inputs into simpler components, while time invariance ensures that the same operation is applied regardless of when an input occurs. Together, these properties lead to convolution, impulse-response analysis, frequency-response analysis, transfer functions, pole-zero descriptions, and state-space models.

This chapter develops these tools as a connected framework. The impulse response provides a complete description of an LTI system under appropriate conditions. Convolution determines the output for an arbitrary input. Difference equations describe implementable systems, while poles and zeros reveal dynamic behavior, resonance, attenuation, and stability. State-space representations provide an alternative description particularly valuable for high-order systems, control applications, and numerical implementation.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Define and test linearity and time invariance for discrete-time systems.

2. Derive and interpret the impulse response and convolution sum.

3. Distinguish causal, noncausal, stable, and memoryless systems.

4. Analyze systems described by linear constant-coefficient difference equations.

5. Relate impulse response, frequency response, transfer function, poles, and zeros.

6. Determine stability and causality from impulse responses and pole locations.

7. Construct and interpret direct-form and state-space realizations.

8. Apply LTI analysis to engineering design and troubleshooting problems.

3.1 What Makes a System Linear and Time Invariant?

Let a system be denoted by

\[y[n]=\mathcal{T}\{x[n]\}.\]

It is linear if it satisfies both additivity and homogeneity. Equivalently, for arbitrary sequences \(x_1[n]\), \(x_2[n]\), and constants \(a\), \(b\),

\[\mathcal{T}\{a x_1[n]+b x_2[n]\}=a y_1[n]+b y_2[n].\]

A system is time invariant if a delay in the input produces an identical delay in the output. If \(x[n]\) produces \(y[n]\), then

\[x[n-n_0]\longrightarrow y[n-n_0]\]

for every integer shift \(n_0\). A system can be linear but time varying, or time invariant but nonlinear; both properties are required for an LTI system.

For example, \(y[n]=3x[n]-x[n-1]\) is linear and time invariant. In contrast, \(y[n]=n x[n]\) is linear but time varying, and \(y[n]=x^2[n]\) is time invariant but nonlinear. The operation \(y[n]=x[n]+2\) is neither linear nor, under the usual zero-input definition, suitable as a linear system because a zero input does not produce a zero output.

The importance of LTI systems follows from the representation of any sequence as a weighted sum of shifted unit impulses:

\[x[n]=\sum_{k=-\infty}^{\infty}x[k]\delta[n-k].\]

Here, \(\delta[n]=1\) for \(n=0\) and is zero otherwise. If the response to \(\delta[n]\) is \(h[n]\), time invariance gives the response to \(\delta[n-k]\) as \(h[n-k]\). By linearity, the response to the complete input is the sum of these shifted responses.

3.2 Impulse Response and the Convolution Sum

The impulse response of an LTI system is

\[h[n]=\mathcal{T}\{\delta[n]\}.\]

Substitution of the impulse expansion of the input gives

\[

y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k].

\]

This operation is the discrete-time convolution, written

\[y[n]=x[n]*h[n].\]

An equivalent form is

\[y[n]=\sum_{k=-\infty}^{\infty}h[k]x[n-k].\]

Convolution is commutative, associative, and distributive:

\[xh=hx,\qquad (xh)g=x(hg),\]

\[x(h+g)=xh+x*g.\]

These properties have practical significance. Commutativity means that a cascade of two LTI systems can be analyzed in either order. Associativity permits several cascaded blocks to be replaced by one equivalent system whose impulse response is the convolution of the individual responses. Distributivity describes parallel connections.

Graphical and computational interpretation

To compute \(x[n]*h[n]\), one sequence is reversed, shifted, multiplied point by point by the other, and summed. In numerical implementation, direct convolution requires approximately \(NM\) multiplications for sequences of lengths \(N\) and \(M\). For long sequences, fast convolution based on the discrete Fourier transform can be more efficient, although block processing introduces latency and boundary-management issues.

Consider

\[x[n]=\{1,2,1\},\qquad h[n]=\{1,-1\},\]

with both sequences beginning at \(n=0\). The output is

\[y[0]=1,\quad y[1]=2-1=1,\quad y[2]=1-2=-1,\quad y[3]=-1.\]

Thus \(y[n]=\{1,1,-1,-1\}\). The example illustrates that convolution combines overlapping samples rather than simply multiplying corresponding samples.

3.3 Causality, Memory, and Stability

A discrete-time system is causal if the output at time \(n_0\) depends only on present and past inputs, not on future values. For an LTI system, causality is equivalent to

\[h[n]=0\quad \text{for }n<0.\]

A causal LTI system can therefore be implemented in real time, provided its delays and computations are physically realizable. A noncausal system may still be useful in offline processing, such as zero-phase filtering of a stored data record.

A system is memoryless if \(y[n]\) depends only on \(x[n]\). For an LTI system, this occurs when

\[h[n]=K\delta[n],\]

so that \(y[n]=Kx[n]\). Any dependence on delayed or advanced input samples represents memory.

BIBO stability means that every bounded input produces a bounded output. If \(|x[n]|\leq B_x\), then an LTI system is BIBO stable if

\[\sum_{n=-\infty}^{\infty}|h[n]|<\infty.\]

This condition is both necessary and sufficient for discrete-time LTI systems. It is called absolute summability. A bounded impulse response is not sufficient: for example, \(h[n]=1\) for all nonnegative \(n\) is bounded but not absolutely summable, and a bounded constant input produces an unbounded output.

For a causal rational system, stability can also be tested using poles: every pole must lie strictly inside the unit circle in the complex \(z\)-plane. Poles on or outside the unit circle indicate instability or marginal cases requiring careful interpretation.

3.4 Difference-Equation Models

Many digital systems are specified by a linear constant-coefficient difference equation:

\[

\sum_{k=0}^{N}a_k y[n-k]=\sum_{k=0}^{M}b_k x[n-k].

\]

Usually \(a_0=1\), giving

\[y[n]=-\sum_{k=1}^{N}a_k y[n-k]+\sum_{k=0}^{M}b_k x[n-k].\]

The terms involving previous outputs constitute feedback and create an infinite impulse response (IIR) system in general. If there are no previous-output terms, the system is a finite impulse response (FIR) system:

\[y[n]=\sum_{k=0}^{M}b_kx[n-k].\]

For an initially relaxed system, taking the z-transform yields the system function

\[H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{k=0}^{M}b_kz^{-k}}{1+\sum_{k=1}^{N}a_kz^{-k}}.\]

The numerator determines zeros, and the denominator determines poles. The transfer-function description is valid with a specified region of convergence and initial conditions. Nonzero initial conditions produce an additional zero-input response, so the total response is the sum of zero-state and zero-input responses.

FIR and IIR trade-offs

FIR filters are naturally causal when implemented with delays, are always BIBO stable for finite coefficients, and can possess exactly linear phase when their coefficients have appropriate symmetry. Their main cost is often a higher order for a given magnitude specification. IIR filters can achieve sharp selectivity with fewer storage elements and multiplications, but feedback makes them more sensitive to coefficient quantization, overflow, limit cycles, and stability loss.

3.5 Frequency Response and Eigenfunction Interpretation

Complex exponentials are eigenfunctions of LTI systems. Let

\[x[n]=e^{j\omega n}.\]

Then

\[

y[n]=\sum_k h[k]e^{j\omega(n-k)}

=e^{j\omega n}\sum_k h[k]e^{-j\omega k}.

\]

Define the frequency response

\[H(e^{j\omega})=\sum_{k=-\infty}^{\infty}h[k]e^{-j\omega k}.\]

Therefore,

\[y[n]=H(e^{j\omega})e^{j\omega n}.\]

The magnitude \(|H(e^{j\omega})|\) gives the amplitude scaling, while the phase \(\arg H(e^{j\omega})\) gives phase shift. Since discrete-time frequency response is periodic with period \(2\pi\), frequencies separated by integer multiples of \(2\pi\) are equivalent.

For a real-valued impulse response,

\[H(e^{-j\omega})=H^*(e^{j\omega}),\]

so the magnitude is even and the phase has corresponding odd symmetry, apart from phase-unwrapping conventions.

The frequency response is the transfer function evaluated on the unit circle:

\[H(e^{j\omega})=H(z)|_{z=e^{j\omega}},\]

provided the unit circle lies in the region of convergence. This qualification is essential for systems whose algebraic transfer function is the same but whose causal and anti-causal realizations have different stability properties.

3.6 Poles, Zeros, and the z-Plane

A zero is a value of \(z\) that makes the numerator of \(H(z)\) zero. A pole is a value that makes the denominator zero after cancellation is excluded. Zeros suppress selected complex-exponential components, while poles amplify responses near their angular locations and control natural modes.

For a causal rational system, a pole at \(p\) contributes a mode proportional to \(p^n u[n]\). Its magnitude is governed by \(|p|^n\): poles inside the unit circle decay, poles outside grow, and poles on the unit circle persist without decay. Complex-conjugate poles generate damped oscillations in real-coefficient systems.

A zero near the unit circle produces attenuation near its angle. A pole near the unit circle produces a narrow resonant peak near its angle. These facts allow engineers to interpret frequency-selective behavior directly from a pole-zero plot. However, pole-zero proximity can cause severe sensitivity: nearly cancelling a pole with a zero may produce a desired nominal response but poor robustness under finite-precision arithmetic.

3.7 State-Space Representation

A state-space model expresses a system using internal variables. In discrete time,

\[

\mathbf{x}[n+1]=\mathbf{A}\mathbf{x}[n]+\mathbf{B}x[n],

\]

\[

y[n]=\mathbf{C}\mathbf{x}[n]+D x[n].

\]

The vector \(\mathbf{x}[n]\) is the state, and matrices \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), and \(D\) describe state evolution, input coupling, output observation, and direct feedthrough. With zero initial state,

\[H(z)=\mathbf{C}(z\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}+D.\]

The eigenvalues of \(\mathbf{A}\) are the system poles, subject to pole-zero cancellations and realization issues. State-space form is especially useful for high-order systems, multivariable systems, time-varying extensions, control design, and numerical algorithms. Different state choices can represent the same input-output system; such realizations are related by nonsingular coordinate transformations.

A realization should be controllable if every state can be influenced by the input and observable if every state can affect the output. Uncontrollable or unobservable modes may exist in a nonminimal realization and can complicate diagnosis, estimation, and implementation.

3.8 A Unified Worked Example

Consider the causal system

\[y[n]-0.6y[n-1]=x[n].\]

With initial rest,

\[H(z)=\frac{1}{1-0.6z^{-1}}.\]

The pole is at \(z=0.6\), so the causal system is stable. Expanding the denominator gives

\[H(z)=\sum_{k=0}^{\infty}(0.6)^kz^{-k},\]

and therefore

\[h[n]=(0.6)^n u[n].\]

The impulse response is causal and absolutely summable because

\[\sum_{n=0}^{\infty}|0.6|^n=\frac{1}{1-0.6}=2.5.\]

Its frequency response is

\[H(e^{j\omega})=\frac{1}{1-0.6e^{-j\omega}}.\]

The magnitude is largest near \(\omega=0\), so the system behaves as a low-pass recursive filter. For a unit-step input, the output approaches the steady value

\[\lim_{n\to\infty}y[n]=\frac{1}{1-0.6}=2.5.\]

The same algebraic expression could be associated with an anti-causal region of convergence \(|z|<0.6\), but that realization would not be causal. This example shows why a transfer-function formula alone is incomplete without causality, initial-condition assumptions, and region of convergence.

Applied Case Study: Vibration Monitoring in an Industrial Motor

Background

An industrial motor is monitored using an accelerometer sampled at a fixed rate. Mechanical imbalance produces a strong component at the shaft rotational frequency, while bearing defects generate higher-frequency components. The monitoring algorithm must suppress broadband measurement noise while retaining narrowband fault indicators.

Situation and problem

An engineer initially selects a high-order recursive filter with poles placed close to the unit circle to obtain sharp spectral selectivity. In floating-point simulation, the filter appears satisfactory. On the embedded processor, however, occasional large transients and persistent low-level oscillations occur after overload events.

Analysis

The pole-zero analysis identifies two related risks. First, poles near the unit circle produce long settling times and amplify round-off effects. Second, finite-word-length arithmetic can perturb pole locations outward, converting a nominally stable system into an unstable or practically unusable one. The direct-form implementation also has large internal intermediate values even though the final output remains within range.

Approach and solution

The design team replaces the single high-order section with a cascade of second-order sections, scales each section to control internal signal levels, and verifies pole radii under coefficient quantization. The impulse response is examined to measure settling time, while the frequency response is checked at the shaft and fault frequencies. A parallel FIR diagnostic channel is added for transient detection because its predictable finite response is easier to validate.

Outcome

The revised implementation preserves the required attenuation and detection bandwidth while reducing overflow and numerical sensitivity. The system also includes startup handling, saturation monitoring, and recorded diagnostic states so that a maintenance engineer can distinguish a mechanical fault from a digital implementation fault.

Learning points

The case illustrates that LTI analysis is not limited to symbolic manipulation. Pole radius affects transient duration, realization structure affects numerical reliability, and stability must be evaluated for the implemented coefficients rather than only for ideal design values. Engineering decisions therefore require simultaneous consideration of frequency response, time response, hardware arithmetic, and diagnostic requirements.

Practical Design, Risks, and Professional Considerations

A reliable LTI design begins with a precise specification: sampling rate, passbands, stopbands, allowable delay, transient requirements, arithmetic precision, and expected signal amplitude. Causality and latency must be addressed before selecting an ideal frequency response. Offline processing may permit zero-phase forward-backward filtering, but such processing is noncausal in real time and changes endpoint behavior.

Common mistakes include confusing boundedness of \(h[n]\) with absolute summability, evaluating a transfer function on the unit circle without checking the region of convergence, ignoring initial conditions, and assuming that pole-zero cancellation removes all practical effects. Exact cancellation may fail under coefficient quantization, and hidden unstable modes can remain in nonminimal realizations.

In safety-critical, medical, automotive, and industrial applications, filter behavior should be validated using requirements traceability, independent test signals, worst-case numerical analysis, and documented version control. A filter that is mathematically stable may still be unsafe if delay, saturation, missing samples, or abnormal startup conditions are ignored. Professional practice also requires transparent reporting of assumptions, sampling limitations, and performance under out-of-range inputs.

Modern design tools automate pole-zero placement, frequency-response optimization, code generation, and fixed-point analysis. Automated tools are valuable, but they do not replace interpretation. Engineers must inspect the generated structure, scaling, quantization sensitivity, latency, and behavior under exceptional conditions. Data-driven or adaptive methods may change coefficients over time; once coefficients vary, the system is generally time varying and ordinary fixed LTI conclusions must be applied with care.

Best-Practice Checklist

• State the input-output model and initial-condition assumptions.
• Test linearity and time invariance before using convolution or frequency-response results.
• Use impulse response to verify causality, memory, settling, and stability.
• Check poles, zeros, and the region of convergence together.
• Evaluate frequency response on the unit circle only when justified.
• Distinguish FIR stability from the numerical stability of its implementation.
• For IIR systems, use suitable cascaded sections and coefficient scaling.
• Analyze quantization, overflow, limit cycles, and startup transients.
• Validate both nominal performance and worst-case implementation behavior.

Chapter Summary

A discrete-time LTI system is characterized by linearity and time invariance. Its impulse response completely determines its zero-state input-output behavior, and convolution provides the output for any input sequence. Causality corresponds to a right-sided impulse response, while BIBO stability requires absolute summability. Difference equations lead to transfer functions whose poles and zeros describe dynamic modes and frequency selectivity. For causal rational systems, poles strictly inside the unit circle ensure stability. Frequency response is obtained by evaluating the transfer function on the unit circle when the region of convergence permits it. State-space models describe internal dynamics and are particularly useful for high-order and multivariable systems. Practical design must also address finite precision, implementation structure, latency, saturation, and validation under abnormal conditions.

Key Terms

Impulse response: Output of a system produced by a unit impulse input.

LTI system: A system possessing both linearity and time invariance.

Convolution: Operation that combines an input sequence with an impulse response to obtain the output.

Causality: Property that the current output does not depend on future input samples.

Memoryless system: System whose output depends only on the input at the same time index.

BIBO stability: Property that every bounded input produces a bounded output.

Absolute summability: Finiteness of the sum of the absolute values of an impulse response.

FIR system: System with a finite-duration impulse response and generally no feedback.

IIR system: System with an impulse response of potentially infinite duration, usually due to feedback.

Difference equation: Discrete-time relation connecting input and output samples.

Transfer function: Ratio \(Y(z)/X(z)\) under zero initial conditions.

Frequency response: Complex gain applied by an LTI system to a complex exponential.

Pole: A value of \(z\) at which a rational transfer function becomes unbounded.

Zero: A value of \(z\) at which a transfer function becomes zero.

Region of convergence: Set of z-plane values for which a z-transform converges.

State-space representation: Model using state-update and output equations.

Controllability: Ability of an input to influence all states of a realization.

Observability: Ability to infer or detect states from the output.

Unit circle: Locus \(|z|=1\) in the complex z-plane.

Linear phase: Phase response that varies linearly with frequency over a specified range.

Review Questions

Short Answer Questions

1. State the linearity and time-invariance conditions for a discrete-time system.

2. Define the impulse response of an LTI system.

3. What condition on \(h[n]\) guarantees BIBO stability?

4. Distinguish FIR and IIR systems.

5. Why must the region of convergence be considered with a transfer function?

Long Answer Questions

1. Derive the convolution sum from the representation of a sequence as shifted impulses.

2. Explain how causality and stability can be determined from an LTI impulse response.

3. Derive the transfer function of a linear constant-coefficient difference equation.

4. Explain the relationship among frequency response, poles, zeros, and the unit circle.

5. Describe the state-space representation of a discrete-time LTI system and explain controllability and observability.

Analytical / Application Questions

1. For \(y[n]-0.8y[n-1]=x[n]\), determine the pole, causal impulse response, stability, and steady-state output for a unit-step input.

2. A filter has poles at \(0.98e^{\pm j0.4}\). Predict its transient duration, frequency-selective behavior, and sensitivity to coefficient quantization.

3. An offline signal-processing system uses a symmetric noncausal impulse response to obtain zero phase distortion. Explain why this is unsuitable for real-time operation and propose a practical alternative.

22. Multiple Choice Questions with Answers

Question 1

Which condition is sufficient and necessary for BIBO stability of a discrete-time LTI system?

A. The impulse response is bounded

B. The impulse response is absolutely summable

C. All zeros lie inside the unit circle

D. The system is memoryless

Correct Answer: B. The impulse response is absolutely summable

Question 2

For an LTI system, causality requires that the impulse response satisfy

A. h[n]=0 for n>0

B. h[n]=0 for n=0

C. h[n]=0 for n<0

D. h[n] be periodic

Correct Answer: C. h[n]=0 for n<0

Question 3

The output of an LTI system for an arbitrary input x[n] is obtained from

A. Addition of x[n] and h[n]

B. Convolution of x[n] and h[n]

C. Division of x[n] by h[n]

D. Time reversal of x[n] only

Correct Answer: B. Convolution of x[n] and h[n]

Question 4

For the causal system H(z)=1/(1−0.6z⁻¹), the impulse response is

A. (0.6)ⁿu[n]

B. (−0.6)ⁿu[n]

C. (0.6)ⁿu[−n]

D. δ[n−0.6]

Correct Answer: A. (0.6)ⁿu[n]

Question 5

A causal rational LTI system is BIBO stable when

A. All poles lie strictly inside the unit circle

B. All zeros lie strictly inside the unit circle

C. At least one pole lies on the unit circle

D. All poles lie outside the unit circle

Correct Answer: A. All poles lie strictly inside the unit circle

Question 6

What does a zero located near the unit circle at angle ω₀ generally do?

A. Creates amplification near ω₀

B. Attenuates frequencies near ω₀

C. Makes the system necessarily noncausal

D. Eliminates all system memory

Correct Answer: B. Attenuates frequencies near ω₀

Question 7

Which statement about complex exponentials and LTI systems is correct?

A. They are generally converted into impulses

B. They are eigenfunctions, scaled by H(e^{jω})

C. They always produce zero output

D. They destroy time invariance

Correct Answer: B. They are eigenfunctions, scaled by H(e^{jω})

Question 8

In the difference equation y[n]=0.7y[n−1]+x[n], feedback produces

A. An FIR system of length one

B. An IIR system in general

C. A memoryless system

D. A non-linear system

Correct Answer: B. An IIR system in general

Question 9

In a state-space model, the eigenvalues of matrix A generally correspond to

A. System poles

B. System input samples

C. Zeros only

D. The sampling frequency

Correct Answer: A. System poles

Question 10

Why can a nominally stable high-order IIR filter become problematic in fixed-point hardware?

A. Quantization can perturb poles and cause overflow or instability

B. FIR filters always have feedback

C. The unit circle disappears in fixed-point arithmetic

D. Convolution cannot be implemented digitally

Correct Answer: A. Quantization can perturb poles and cause overflow or instability