Week 8 Assignment: Digital Filter Structures 1. The Figure S ✓ Solved

Week8 Assignment: Digital Filter Structures 1. The figure sh

Week8 Assignment: Digital Filter Structures 1. The figure shown below is a typical closed-loop discrete-time feedback control system in which G(z) is the plant and C(z) is the compensator. If C(z) = K, determine the overall transfer function.

2. Develop a canonic direct-form II realization of the transfer function.

3. Realize the transfer function in parallel form.

4. Consider a length-8 FIR filter transfer function H(z) given by ... Develop a four-branch polyphase realization of H(z).

5. Realize the FIR filter transfer function in cascade of three 2nd-order sections.

Assignment 2: z-Transform

1. Find the z-transform and the ROC of the sequence x[n] = (0.5)^n u[n].

2. Find the poles and zeros of the z-transform. Does this function correspond to a convergent sequence?

3. The impulse response of an LTI discrete-time system is h[n] = (0.2)^n u[n]. If the input to this system is x[n] = (0.3)^n u[n], calculate the output using z-transform method.

4. Consider the causal LTI discrete-time system described by y[n] = 3x[n] + 0.4y[n-1] + 0.05y[n-2]. Determine its transfer function.

Assignment 3: Discrete-time Fourier Transform

1. Find the DTFT of the sequence.

2. Determine the DTFT of x[n] = n α^n u[n], with |α|

Assignment 4: Discrete Fourier Transform

1. If the convolution of two periodic sequences with period N is performed, the result is periodic with period N.

2. Determine the 5-point DFT of a given sequence.

3. Show that the circular convolution is commutative and associative.

4. Compute the circular convolution of two length-4 sequences using a matrix equation.

5. Let X[k] be the N-point DFT of x[n]. Find the time-domain sequence whose DFT is X[k] shifted in index.

Assignment 5: IIR Filter Design

1. Determine the peak ripple values and passband/stopband specifications for a given design case.

2. Use bilinear transformation to obtain the digital filter transfer function corresponding to a given analog prototype. Use a specified sampling interval.

3. Design a Butterworth lowpass IIR digital filter of a specified order and cutoff.

4. Starting from the prototype, design a bandstop digital filter using a spectral transformation in the z-domain.

Assignment 6: FIR Filter Design

1. Find the impulse response of a digital notch filter with a given frequency response specification.

2. Determine the smallest length for a lowpass FIR filter to meet given passband/stopband specs.

3. Design a bandpass FIR filter with the smallest length using a fixed-window method under specified specs.

4. Design a lowpass FIR filter with Kaiser window given passband/stopband specs; show the ideal impulse response, filter order, and Kaiser window parameters.

Assignment 1: Discrete-time Signals & Systems

1. Determine whether a given sequence is absolutely summable.

2. Find the fundamental period of x[n] = cos(0.125π n).

3. For finite-length sequences, determine the length of their convolution result.

4. Compute the linear convolution of x[n] with itself for a simple finite sequence.

5. Determine whether a system described by a given difference equation is causal.

Note: You may use MATLAB to solve the problems where necessary. All questions carry equal points. The above prompts cover a structured set of topics in digital filter structures, z-transform analysis, discrete-time Fourier analysis, DFT, and filter design (IIR and FIR) commonly addressed in graduate-level DSP coursework. The cleaned prompt consolidates core tasks across the Week 8 material, focusing on canonical realizations, polyphase implementations, and standard transform techniques.

Paper For Above Instructions

Digital filter structures form a foundational pillar of modern signal processing, enabling efficient realization of both IIR and FIR systems in hardware and software. Across the Week 8 prompts, there is a coherent throughline: from system modeling and transfer-function derivation to concrete realizations in canonical and polyphase forms, and finally to transform-domain analysis and design procedures. A careful treatment of these topics reveals how theory translates into hardware-efficient, numerically robust implementations while preserving the essential magnitude and phase characteristics of the intended filters. This paper presents a cohesive discussion of the cleaned assignment prompts, grounded in standard DSP theory and practical design considerations, with citations to canonical references in the field.

First, the closed-loop feedback description G(z) for the plant and C(z) for the compensator leads to the standard closed-loop transfer function under negative feedback: T(z) = G(z)C(z) / (1 + G(z)C(z)). When the compensator is chosen as a constant gain C(z) = K, the closed-loop transfer function simplifies to T(z) = K G(z) / (1 + K G(z)). This expression highlights the key trade-off in control and filter design: increasing K boosts the forward path gain but also narrows the closed-loop stability and bandwidth margins through the denominator 1 + K G(z). This classical result is discussed in foundational texts on discrete-time control and DSP theory (Oppenheim & Schafer, 2010).

Canonical direct-form II realization is a compact, memory-efficient way to implement a higher-order transfer function H(z) with fewer delay elements than the direct-form I structure. In direct-form II, the filter’s numerator and denominator polynomials share a common delay-line structure, producing a minimal-state realization. The approach reduces implementation cost, particularly for high-order filters, and is widely discussed alongside direct-form I and transposed forms in standard DSP references (Proakis & Manolakis, 2007; Oppenheim & Schafer, 2010).

Realizing a transfer function in parallel form typically involves partial-fraction decomposition or spectral factorization to express H(z) as a sum of simpler, possibly second-order, blocks in parallel. This approach is advantageous for fixed-point implementations, enabling separate scaling and rounding for each branch and improving numerical stability. The parallel form is often used in conjunction with cascade or polyphase decompositions to balance hardware resources and numerical accuracy (Lathi, 2005).

The four-branch polyphase realization of an 8-tap FIR filter H(z) is a practical technique from multirate signal processing. Polyphase decomposition splits the impulse response into multiple sub-filters with decimation to reduce computational load by exploiting structure in the filter taps. For an 8-tap FIR with M = 4 branches, one can express H(z) as H(z) = E0(z^4) + z^{-1}E1(z^4) + z^{-2}E2(z^4) + z^{-3}E3(z^4), where Ek(z) are the polyphase components. This decomposition reduces cost on hardware and is the basis of efficient implementations in polyphase filter banks and multirate systems (Vaidyanathan, 1993).

Cascade-of-second-order-sections (SOS) realizations are a standard technique for robustly implementing high-order FIR and IIR filters, particularly in fixed-point environments where numerical stability is critical. By decomposing a higher-order transfer function into a cascade of second-order blocks, one can carefully manage pole placement, damping, and overall numerical sensitivity. This realization is a near-universal approach in modern DSP toolchains (Proakis & Manolakis, 2007).

The z-transform problems in Assignment 2 emphasize core concepts: calculating Z{(0.5)^n u[n]} and its ROC, identifying poles and zeros to assess convergence, and deriving Y(z)/X(z) for IIR systems like y[n] = 3x[n] + 0.4y[n-1] + 0.05y[n-2]. The unilateral/causal sequence X[n] = (0.5)^n u[n] yields X(z) = 1 / (1 − 0.5 z^{-1}) with ROC |z| > 0.5, while the corresponding IIR transfer function H(z) = 3 / (1 − 0.4 z^{-1} − 0.05 z^{-2}) follows from standard Z-transform algebra. These steps illustrate how transform-domain methods translate time-domain recursions into algebraic forms suitable for stability and frequency response analysis (Oppenheim & Schafer, 2010; Proakis & Manolakis, 2007).

Discrete-time Fourier Transform (DTFT) tasks further connect time-domain sequences to continuous-frequency-domain representations. For a causal exponential sequence x[n] = a^n u[n] with |a|

The Discrete Fourier Transform (DFT) problems emphasize periodicity, circular convolution, and the relationships between time-domain and frequency-domain representations. The DFT requires finite-length sequences and yields an N-point spectrum X[k], with the inverse transforming back to x[n]. The circular convolution property is essential in efficiently computing linear convolutions via the DFT, motivating the exploration of multiplication in the frequency domain as a tool for O(N log N) implementations using FFTs (Mitra, 2001; Oppenheim & Schafer, 2010).

Finally, the FIR and IIR design prompts address practical constraints such as ripple, passband/stopband attenuation, windowing (e.g., Kaiser window), bilinear transformation for analog-to-digital conversion, and spectral transformations for bandstop and bandpass design (Kaiser window parameters are chosen based on desired ripple and attenuation; bilinear transform preserves stability while warping frequency, which must be compensated in the design). The literature provides clear methodologies for these steps, including determining filter order with Kaiser’s formula, selecting window parameters, and implementing fixed-window or minimum-multiplier optimizations for real-time systems (Oppenheim & Schafer, 2010; Mitra, 2001; Kuo et al., 1998; Vaidyanathan, 1993).

In sum, the Week 8 prompts solicit a comprehensive, practice-oriented engagement with canonical filter realizations (direct-form II, parallel, and polyphase), standard transforms (Z-transform, DTFT, DFT), and practical design methods (IIR/FIR design, Kaiser window, bilinear transform). The expected deliverables span theoretical derivations, structural realizations, and design methodologies, all of which are well-established in the DSP literature and are essential for building robust, efficient digital filters and controllers.

References

• Oppenheim, A. V., Schafer, R. W. (2010). Discrete-Time Signal Processing. 3rd ed. Prentice Hall.
• Proakis, J. G., Manolakis, D. G. (2007). Digital Signal Processing: Principles, Algorithms, and Applications. 4th ed. Prentice-Hall.
• Lyons, R. G. (2010). Understanding Digital Signal Processing. 3rd ed. Pearson.
• Rabiner, L. R., Gold, B. (1986). Theory and Applications of Digital Signal Processing. Prentice-Hall.
• Mitra, S. K. (2001). Digital Signal Processing: A Computer-Based Approach. 2nd ed. McGraw-Hill.
• Lathi, B. P. (2005). Linear Systems and Signals. Oxford University Press.
• Oppenheim, A. V., Willsky, A. S., Nawab, S. H. (1997). Signals and Systems. Prentice-Hall.
• Vaidyanathan, P. P. (1993). Multirate Systems and Filter Banks. Prentice-Hall.
• Smith, S. W. (1997). The Scientist and Engineer's Guide to Digital Signal Processing. California Tech.
• Haykin, S. (2002). Adaptive Filter Theory. 4th ed. Prentice Hall.