{"pinia":{"ui":{"prefs":{"theme":"system","textScale":"md","fontFamily":"sans","reducedMotion":false,"highContrast":false}},"academy":{"universities":[],"universitySummaries":[{"slug":"rtu","name":"RTU","fullName":"Rajasthan Technical University","branches":[{"slug":"first-year-common","name":"First Year (Common)"},{"slug":"computer-science","name":"Computer Science"},{"slug":"mechanical","name":"Mechanical"},{"slug":"civil","name":"Civil"},{"slug":"ai-data-science","name":"AI & Data Science"},{"slug":"ee-ec-eee","name":"EE / EC / EEE"}]},{"slug":"aktu","name":"AKTU","fullName":"Dr. A. P. J. Abdul Kalam Technical University","branches":[{"slug":"first-year-common","name":"First Year (Common)"},{"slug":"computer-science","name":"Computer Science"}]}],"activePapers":[{"id":"ee-ec-eee-signals-and-systems-2023","title":"Signals and Systems (2023)","subjectSlug":"signals-and-systems","subjectName":"Signals and Systems","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":4,"examYear":2023,"paperCode":"4E1323","questions":[{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Differentiate between Even and odd signals."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define the unit impulse sequence."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What do you mean by the Causal and Non-causal systems."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the condition for stability of system."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Differentiate continuous - time and discrete - time systems."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What are linear constant - coefficient differential equations."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define Region of convergence (ROC) for Laplace transform."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the Laplace transform of unit impulse function."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define interpolation."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State the formula for Fourier transform."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"State and explain the Parseval's theorem."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the orthogonality of the signals $\\sin(\\omega t)$ and $\\sin(2\\omega t)$ over the time interval $(0,T)$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Compute the output y(t) for a continuous - time LTI system whose impulse response h(t) and the input x(t) are given by $h(t) = e^{-at} u(t)$ and $x(t) = e^{at} u(-t)$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the minimum sampling frequency for non - zero frequency spectra $X(j\\omega)$ defined between $10\\text{kHz} < |\\omega| < 20\\text{kHz}$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"What are energy and power signals?"}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Explain the properties of Laplace transform."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partb-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"What is zero - order hold, derive the expression for transfer function of zero - order hold?"}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"The discrete - time system shown in Fig. 1 is known as unit delay element. Determine whether system is (a) memoryless (b) causal (c) linear (d) time - invariant (e) stable.\n\n$$\\text{x[n]} \\longrightarrow \\boxed{\\text{Unit Delay}} \\longrightarrow \\text{y[n] = x[n-1]}$$\n\n$$\\text{Fig. 1 Unit Delay Element}$$"}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Find the Z-transform X(z) and sketch the pole zero plot with the ROC for $x[n] = \\left(\\frac{1}{3}\\right)^n u[n] + \\left(\\frac{1}{2}\\right)^n u[-n-1]$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Consider a continuous - time LTI system described by\n\n$$\\frac{dy(t)}{dt} + 2y(t) = x(t)$$\n\nusing fourier transform, find the output y(t) to the input $x(t) = u(t)$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Describe the various methods of reconstruction of signals from its samples."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2023-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Compute $y[n] = x[n] * h[n]$, where $x[n] = a^n u[n]$, $h[n] = a^{-n} u[-n]$, $0 < a < 1$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}}]},{"id":"ee-ec-eee-signals-and-systems-2024","title":"Signals and Systems (2024)","subjectSlug":"signals-and-systems","subjectName":"Signals and Systems","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":4,"examYear":2024,"paperCode":"4E1323","questions":[{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define signal. Differentiate between discrete time signal and continuous time signal."}],"answer":{"summary":"A signal conveys information about a physical phenomenon. CT signals are defined at every instant, while DT signals are defined only at discrete steps.","explanation":[{"type":"paragraph","content":"**Signal:** A signal is any physical quantity that varies with respect to time, space, or any other independent variable, and carries information about the state or behavior of a physical system."},{"type":"paragraph","content":"- **Continuous-Time (CT) Signal:** Defined for every continuous instant of time. Its domain is the set of real numbers (e.g., $x(t)$)."},{"type":"paragraph","content":"- **Discrete-Time (DT) Signal:** Defined only at specific, discrete intervals of time. Its domain is the set of integers (e.g., $x[n]$)."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Discuss the terms:\n\n(i) Linearity\n(ii) Causality"}],"answer":{"summary":"Linearity implies the system obeys superposition. Causality implies the output depends only on past and present inputs.","explanation":[{"type":"paragraph","content":"**(i) Linearity:** A system is linear if it satisfies the principle of superposition, which includes both additivity and homogeneity (scaling). If input $x_1(t)$ yields $y_1(t)$ and $x_2(t)$ yields $y_2(t)$, then $a x_1(t) + b x_2(t)$ must yield $a y_1(t) + b y_2(t)$."},{"type":"paragraph","content":"**(ii) Causality:** A system is causal if its output at any given time depends entirely on present and past inputs, and not on future inputs. All real-world physical systems are causal."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Discuss the relationship between z-transform and DTFT."}],"answer":{"summary":"The DTFT is a special case of the Z-transform evaluated along the unit circle in the z-plane.","explanation":[{"type":"paragraph","content":"The Z-transform of a sequence $x[n]$ is $X(z) = \\sum x[n]z^{-n}$. The Discrete-Time Fourier Transform (DTFT) is $X(e^{j\\omega}) = \\sum x[n]e^{-j\\omega n}$."},{"type":"paragraph","content":"The relationship is that the **DTFT is simply the Z-transform evaluated along the unit circle** in the complex z-plane. That is, if we substitute $z = e^{j\\omega}$ (where $|z| = 1$), the Z-transform reduces to the DTFT."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"How can aliasing be eliminated? Explain."}],"answer":{"summary":"Aliasing is eliminated by sampling above the Nyquist rate and using an anti-aliasing low-pass filter.","explanation":[{"type":"paragraph","content":"Aliasing (high-frequency components folding into low frequencies) can be eliminated by two main steps:"},{"type":"paragraph","content":"1. **Anti-aliasing Filter:** Passing the continuous signal through a low-pass analog filter before sampling to strictly band-limit the signal and remove frequencies higher than $f_m$."},{"type":"paragraph","content":"2. **Nyquist Rate:** Ensuring the sampling frequency $f_s$ is strictly greater than twice the maximum frequency present in the filtered signal ($f_s > 2f_m$)."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What do you understand by reconstruction?"}],"answer":{"summary":"Reconstruction is the mathematical or physical process of converting a discrete-time sampled signal back into its continuous-time form.","explanation":[{"type":"paragraph","content":"**Reconstruction** is the process of accurately recreating the original continuous-time analog signal from its discrete-time sampled values. This is practically achieved using a Digital-to-Analog Converter (DAC) followed by a low-pass reconstruction (smoothing) filter."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What are the advantages of DFT in comparison of DTFT?"}],"answer":{"summary":"DFT is computationally feasible for digital computers because it is discrete and finite, unlike the continuous DTFT.","explanation":[{"type":"paragraph","content":"The primary advantage of the **Discrete Fourier Transform (DFT)** over the DTFT is that the DFT is discrete and finite in both the time and frequency domains. The DTFT produces a continuous frequency spectrum which requires infinite memory to store. The DFT evaluates a finite number of frequency samples, making it perfectly suited for processing on digital computers, especially utilizing highly efficient Fast Fourier Transform (FFT) algorithms."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define circular convolution."}],"answer":{"summary":"Circular convolution evaluates the convolution of two finite sequences by treating them as periodic.","explanation":[{"type":"paragraph","content":"**Circular Convolution** is a type of convolution used in discrete signal processing where two finite-length sequences of length N are convolved. Unlike linear convolution, it treats the sequences as if they are periodic (circular) with period N. The operation involves shifting one sequence circularly modulo N."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Differentiate between impulse response and step response."}],"answer":{"summary":"Impulse response is the output for an impulse input; step response is the output for a step input.","explanation":[{"type":"paragraph","content":"- **Impulse Response ($h(t)$):** The output of an LTI system when the input is a unit impulse function $\\delta(t)$."},{"type":"paragraph","content":"- **Step Response ($s(t)$):** The output of the system when the input is a unit step function $u(t)$."},{"type":"paragraph","content":"Mathematically, the step response is the integral of the impulse response: $s(t) = \\int_{-\\infty}^{t} h(\\tau) d\\tau$."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Discuss state transition matrix and its role."}],"answer":{"summary":"The state transition matrix describes how the internal state of a system evolves over time with zero input.","explanation":[{"type":"paragraph","content":"The **State Transition Matrix**, denoted by $\\Phi(t) = e^{At}$, is a fundamental matrix in state-space control theory. Its role is to characterize the unforced response of a system. It completely describes how the internal state of a linear continuous system evolves from an initial state $x(0)$ to any time $t > 0$ when the external input is zero."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define magnitude and phase response."}],"answer":{"summary":"Magnitude response is the absolute value of the frequency response; phase response is the angle.","explanation":[{"type":"paragraph","content":"For a system's frequency response $H(j\\omega)$:"},{"type":"paragraph","content":"- **Magnitude Response $|H(j\\omega)|$:** Shows how the system amplifies or attenuates different frequency components."},{"type":"paragraph","content":"- **Phase Response $\\angle H(j\\omega)$:** Shows the phase shift (time delay) introduced by the system to different frequency components."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"For the following input-output relationship, determine whether the corresponding system is linear, time invariant or both.\n\n(a) $y(t) = t^2 x(t-1)$\n(b) $y[n] = x^2[n-2]$"}],"answer":{"summary":"System (a) is Linear but Time-Variant. System (b) is Non-Linear but Time-Invariant.","explanation":[{"type":"paragraph","content":"**(a) $y(t) = t^2 x(t-1)$**"},{"type":"paragraph","content":"- **Linearity:** Let input $a x_1(t) + b x_2(t)$ be applied. The output is $t^2 [a x_1(t-1) + b x_2(t-1)] = a y_1(t) + b y_2(t)$. Thus, it satisfies superposition and is **Linear**."},{"type":"paragraph","content":"- **Time-Invariance:** Delay the input by $t_0$: $x_1(t) = x(t-t_0)$. Output $y_1(t) = t^2 x(t-t_0-1)$. Now delay the original output by $t_0$: $y(t-t_0) = (t-t_0)^2 x(t-t_0-1)$. Since $y_1(t) \\neq y(t-t_0)$, the system is **Time-Variant**."},{"type":"paragraph","content":"**(b) $y[n] = x^2[n-2]$**"},{"type":"paragraph","content":"- **Linearity:** Let input be $c x[n]$. The output is $(c x[n-2])^2 = c^2 x^2[n-2] \\neq c y[n]$. It violates homogeneity, so it is **Non-Linear**."},{"type":"paragraph","content":"- **Time-Invariance:** Delay input by $n_0$: $x_1[n] = x[n-n_0]$. Output $y_1[n] = x_1^2[n-2] = x^2[n-n_0-2]$. Delaying the original output yields $y[n-n_0] = x^2[n-n_0-2]$. Since $y_1[n] = y[n-n_0]$, it is **Time-Invariant**."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"State and prove the initial value and final value theorem for Laplace transform."}],"answer":{"summary":"Initial Value Theorem determines x(0^+). Final Value Theorem determines x(∞) from the Laplace transform.","explanation":[{"type":"heading","content":"Initial Value Theorem (IVT)"},{"type":"paragraph","content":"**Statement:** If $x(t)$ and its derivative are Laplace transformable, then $x(0^+) = \\lim_{s \\to \\infty} s X(s)$."},{"type":"paragraph","content":"**Proof:** Consider the transform of the derivative: $\\mathcal{L}\\{x'(t)\\} = \\int_{0}^{\\infty} x'(t) e^{-st} dt = sX(s) - x(0^+)$.\nTake the limit as $s \\to \\infty$ on both sides. On the left side, as $s \\to \\infty$, $e^{-st} \\to 0$, making the integral zero.\n$0 = \\lim_{s \\to \\infty} [sX(s) - x(0^+)] \\implies x(0^+) = \\lim_{s \\to \\infty} s X(s)$."},{"type":"heading","content":"Final Value Theorem (FVT)"},{"type":"paragraph","content":"**Statement:** If $x(t)$ and its derivative are transformable, and all poles of $sX(s)$ lie in the left half of the s-plane, then $x(\\infty) = \\lim_{s \\to 0} s X(s)$."},{"type":"paragraph","content":"**Proof:** Using the derivative property again: $\\int_{0}^{\\infty} x'(t) e^{-st} dt = sX(s) - x(0^+)$.\nTake the limit as $s \\to 0$ on both sides. On the left side, $e^{-st} \\to 1$.\n$\\int_{0}^{\\infty} x'(t) dt = \\lim_{s \\to 0} [sX(s) - x(0^+)]$\n$[x(\\infty) - x(0^+)] = \\lim_{s \\to 0} sX(s) - x(0^+)$\n$x(\\infty) = \\lim_{s \\to 0} s X(s)$."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the z-transform of the expression $x(n) = a^n u(n) + b^n u(-n-1)$, also draw the ROC."}],"answer":{"summary":"The z-transform is z/(z-a) - z/(z-b) with an annular ROC of |a| < |z| < |b|.","explanation":[{"type":"paragraph","content":"Given $x(n) = a^n u(n) + b^n u(-n-1)$."},{"type":"paragraph","content":"The Z-transform is linear, so we evaluate the terms separately:"},{"type":"paragraph","content":"1. The right-sided sequence $x_1(n) = a^n u(n)$:\n$X_1(z) = \\sum_{n=0}^{\\infty} a^n z^{-n} = \\frac{1}{1 - az^{-1}} = \\frac{z}{z - a}$.\nFor convergence, $|az^{-1}| < 1 \\implies |z| > |a|$."},{"type":"paragraph","content":"2. The left-sided sequence $x_2(n) = b^n u(-n-1)$:\n$X_2(z) = \\sum_{n=-\\infty}^{-1} b^n z^{-n}$. Let $m = -n$.\n$X_2(z) = \\sum_{m=1}^{\\infty} b^{-m} z^{m} = \\sum_{m=1}^{\\infty} (b^{-1}z)^m = \\frac{b^{-1}z}{1 - b^{-1}z} = \\frac{z}{b - z} = \\frac{-z}{z - b}$.\nFor convergence, $|b^{-1}z| < 1 \\implies |z| < |b|$."},{"type":"paragraph","content":"Combining them:"},{"type":"math","content":"X(z) = \\frac{z}{z - a} - \\frac{z}{z - b}"},{"type":"paragraph","content":"**ROC:** The overall ROC is the intersection of the two regions: $|a| < |z| < |b|$. For the ROC to exist, we must have $|a| < |b|$. The ROC visually is a ring (annulus) in the z-plane, strictly bounded between the circles of radius $|a|$ and $|b|$."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Explain sampling of continuous time signal by impulse function with diagram and appropriate mathematical expressions."}],"answer":{"summary":"Ideal sampling multiplies a continuous signal by an impulse train. In the frequency domain, it replicates the spectrum every sampling frequency.","explanation":[{"type":"paragraph","content":"Ideal sampling of a continuous-time signal $x(t)$ is modeled mathematically by multiplying $x(t)$ with a periodic impulse train $p(t)$, also known as a Dirac comb."},{"type":"math","content":"p(t) = \\sum_{n=-\\infty}^{\\infty} \\delta(t - nT_s)"},{"type":"paragraph","content":"Where $T_s$ is the sampling period. The sampled signal $x_p(t)$ is:"},{"type":"math","content":"x_p(t) = x(t) p(t) = \\sum_{n=-\\infty}^{\\infty} x(nT_s) \\delta(t - nT_s)"},{"type":"paragraph","content":"In the frequency domain, multiplication in time becomes convolution in frequency. The Fourier transform of the impulse train is another impulse train in frequency. Thus, the spectrum of the sampled signal $X_p(j\\omega)$ becomes:"},{"type":"math","content":"X_p(j\\omega) = \\frac{1}{T_s} \\sum_{k=-\\infty}^{\\infty} X(j(\\omega - k\\omega_s))"},{"type":"paragraph","content":"This equation shows that the original spectrum $X(j\\omega)$ is scaled by $1/T_s$ and infinitely replicated across the frequency axis at integer multiples of the sampling frequency $\\omega_s = 2\\pi/T_s$. If $\\omega_s > 2\\omega_m$ (Nyquist rate), the replicas do not overlap, and the signal can be reconstructed without aliasing."},{"type":"paragraph","content":"*(A diagram showing a triangle spectrum at origin being replicated at $\\pm \\omega_s, \\pm 2\\omega_s$ is expected here).* "}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the Nyquest rate and discrete signal x(n) for the following continuous time signals:\n\n(a) $x(t) = \\sin(4 \\times 10^3 \\pi t)/\\pi t$\n(b) $x(t) = 1 + \\cos(2000 \\pi t) + \\sin(400 \\pi t)$\n(c) $x(t) = \\text{sin c}(50 \\pi t) + \\text{sin c}(100 \\pi t)$"}],"answer":{"summary":"Nyquist rates are computed by identifying the maximum frequency in each signal and doubling it.","explanation":[{"type":"paragraph","content":"The Nyquist rate is defined as $2 f_{max}$, where $f_{max}$ is the highest frequency component present in the signal."},{"type":"paragraph","content":"**(a) $x(t) = \\sin(4 \\times 10^3 \\pi t)/\\pi t$**\nThis is a sinc function, representing an ideal low-pass filter in the frequency domain with cutoff frequency $\\omega_m = 4000\\pi$ rad/s.\n$2\\pi f_m = 4000\\pi \\implies f_m = 2000$ Hz.\nNyquist rate $= 2 f_m = 2 \\times 2000 = 4000$ Hz."},{"type":"paragraph","content":"**(b) $x(t) = 1 + \\cos(2000 \\pi t) + \\sin(400 \\pi t)$**\nThe signal has three frequency components: $0$ Hz (DC), $1000$ Hz (from $2000\\pi$), and $200$ Hz (from $400\\pi$).\nThe maximum frequency $f_{max} = 1000$ Hz.\nNyquist rate $= 2 f_{max} = 2000$ Hz."},{"type":"paragraph","content":"**(c) $x(t) = \\text{sinc}(50 \\pi t) + \\text{sinc}(100 \\pi t)$**\nAssuming standard unnormalized definition $\\text{sinc}(x) = \\sin(x)/x$. The highest frequency component comes from the second term, $\\omega_m = 100\\pi$ rad/s.\n$2\\pi f_m = 100\\pi \\implies f_{max} = 50$ Hz.\nNyquist rate $= 2 f_{max} = 100$ Hz."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the 4-point DFT of sequence $x(n) = \\cos \\frac{n\\pi}{4}$"}],"answer":{"summary":"The 4-point DFT of the cosine sequence evaluates to X[0]=1, X[1]=1-j\\sqrt{2}, X[2]=1, and X[3]=1+j\\sqrt{2}.","explanation":[{"type":"paragraph","content":"We need the 4-point DFT of $x(n) = \\cos(\\frac{n\\pi}{4})$ for $N=4$ ($n = 0, 1, 2, 3$)."},{"type":"paragraph","content":"Evaluate the sequence $x(n)$:\n$x(0) = \\cos(0) = 1$\n$x(1) = \\cos(\\pi/4) = 1/\\sqrt{2}$\n$x(2) = \\cos(2\\pi/4) = 0$\n$x(3) = \\cos(3\\pi/4) = -1/\\sqrt{2}$"},{"type":"paragraph","content":"The N-point DFT formula is: $X(k) = \\sum_{n=0}^{N-1} x(n) e^{-j \\frac{2\\pi}{N} nk}$"},{"type":"paragraph","content":"For $N=4$, the Twiddle factor is $W_4 = e^{-j\\pi/2} = -j$. Thus, $X(k) = \\sum_{n=0}^{3} x(n) (-j)^{nk}$."},{"type":"paragraph","content":"**For k = 0:**\n$X(0) = 1(1) + \\frac{1}{\\sqrt{2}}(1) + 0(1) - \\frac{1}{\\sqrt{2}}(1) = 1$"},{"type":"paragraph","content":"**For k = 1:**\n$X(1) = 1(1) + \\frac{1}{\\sqrt{2}}(-j)^1 + 0(-j)^2 - \\frac{1}{\\sqrt{2}}(-j)^3 = 1 - j\\frac{1}{\\sqrt{2}} - j\\frac{1}{\\sqrt{2}} = 1 - j\\sqrt{2}$"},{"type":"paragraph","content":"**For k = 2:**\n$X(2) = 1(1) + \\frac{1}{\\sqrt{2}}(-1) + 0(1) - \\frac{1}{\\sqrt{2}}(-1) = 1 - \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{2}} = 1$"},{"type":"paragraph","content":"**For k = 3:**\n$X(3) = 1(1) + \\frac{1}{\\sqrt{2}}(j) + 0(-1) - \\frac{1}{\\sqrt{2}}(-j) = 1 + j\\frac{1}{\\sqrt{2}} + j\\frac{1}{\\sqrt{2}} = 1 + j\\sqrt{2}$"}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partb-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Determine the convolution of the given signals:\n\n$x(t) = u(t-3) - u(t-5)$ and\n$h(t) = e^{-3t} u(t)$\n\nWhere u(t) is unit step function."}],"answer":{"summary":"The convolution integral evaluates piecewise based on the rectangular pulse bounds.","explanation":[{"type":"paragraph","content":"Given signals: $x(t) = u(t-3) - u(t-5)$ (A rectangular pulse from $t=3$ to $t=5$) and $h(t) = e^{-3t} u(t)$."},{"type":"paragraph","content":"The convolution is $y(t) = \\int_{-\\infty}^{\\infty} x(\\tau) h(t - \\tau) d\\tau$."},{"type":"math","content":"y(t) = \\int_{3}^{5} e^{-3(t-\\tau)} u(t-\\tau) d\\tau"},{"type":"paragraph","content":"The condition $u(t-\\tau) = 1$ implies $\\tau \\le t$. We evaluate the integral across three regions of $t$."},{"type":"paragraph","content":"**1. For $t < 3$:**\nThe integration bounds $[3, 5]$ and $\\tau \\le t$ do not overlap. Thus, $y(t) = 0$."},{"type":"paragraph","content":"**2. For $3 \\le t < 5$:**\nThe active integration region is from $3$ to $t$."},{"type":"math","content":"y(t) = \\int_{3}^{t} e^{-3t} e^{3\\tau} d\\tau = e^{-3t} \\left[ \\frac{e^{3\\tau}}{3} \\right]_{3}^{t} = \\frac{e^{-3t}}{3} (e^{3t} - e^{9}) = \\frac{1}{3} (1 - e^{-3(t-3)})"},{"type":"paragraph","content":"**3. For $t \\ge 5$:**\nThe active integration region covers the entire pulse from $3$ to $5$."},{"type":"math","content":"y(t) = \\int_{3}^{5} e^{-3t} e^{3\\tau} d\\tau = e^{-3t} \\left[ \\frac{e^{3\\tau}}{3} \\right]_{3}^{5} = \\frac{e^{-3t}}{3} (e^{15} - e^{9}) = \\frac{1}{3} (e^{-3(t-5)} - e^{-3(t-3)})"}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Describe all the properties of DFT in detail. Also, define the feedback control systems."}],"answer":{"summary":"DFT properties include Linearity, Periodicity, Circular Shift, and Parseval's. Feedback control measures output to reduce system error.","explanation":[{"type":"heading","content":"Properties of Discrete Fourier Transform (DFT)"},{"type":"paragraph","content":"Let $x(n)$ and $y(n)$ be N-point sequences with N-point DFTs $X(k)$ and $Y(k)$."},{"type":"paragraph","content":"1. **Linearity:** $a x(n) + b y(n) \\leftrightarrow a X(k) + b Y(k)$. The DFT of a linear combination of signals is the linear combination of their DFTs."},{"type":"paragraph","content":"2. **Periodicity:** $X(k + N) = X(k)$ and $x(n + N) = x(n)$. Both the time sequence and the frequency spectrum repeat with a period of N."},{"type":"paragraph","content":"3. **Circular Time Shift:** If $x(n)$ is circularly shifted by $m$ samples to $x((n-m))_N$, its DFT becomes $X(k) e^{-j\\frac{2\\pi}{N}mk}$. A circular shift in time introduces a linear phase shift in frequency."},{"type":"paragraph","content":"4. **Circular Convolution:** The circular convolution of two sequences in the time domain equates to the simple multiplication of their DFTs in the frequency domain. $x(n) \\circledast y(n) \\leftrightarrow X(k)Y(k)$."},{"type":"paragraph","content":"5. **Complex Conjugation:** If $x(n)$ is complex, the DFT of $x^*(n)$ is $X^*((-k))_N$. For real sequences, $X(k) = X^*(-k)$, meaning magnitude is even and phase is odd."},{"type":"paragraph","content":"6. **Parseval's Theorem:** $\\sum_{n=0}^{N-1} |x(n)|^2 = \\frac{1}{N} \\sum_{k=0}^{N-1} |X(k)|^2$. The total energy in the discrete-time domain equals the total energy in the discrete-frequency domain."},{"type":"heading","content":"Feedback Control Systems"},{"type":"paragraph","content":"A feedback control system (Closed-Loop System) is one in which the system's output is continuously measured and fed back to be compared with the desired input (reference). The difference between the two (the error signal) is used by a controller to adjust the system's input, effectively reducing the error and forcing the output to track the desired reference. They are highly robust against internal disturbances and parameter variations."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"(a) Find the Laplace transform of following signal $x(t) = e^{-2t} u(-t) + e^{-3t} u(-t)$. Mark the poles and zero of x(s) and also the ROC in the s-plane.\n\n(b) Explain filtering."}],"answer":{"summary":"The signal consists of two left-sided exponentials yielding a ROC of Re(s) < -3. Filtering is selectively altering frequency components.","explanation":[{"type":"heading","content":"(a) Laplace Transform and ROC"},{"type":"paragraph","content":"Given $x(t) = e^{-2t} u(-t) + e^{-3t} u(-t)$. Both terms are left-sided anti-causal sequences."},{"type":"paragraph","content":"Using the standard pair $-e^{-at} u(-t) \\leftrightarrow \\frac{1}{s+a}$ with $ROC: \\text{Re}(s) < -a$:"},{"type":"paragraph","content":"$\\mathcal{L}\\{e^{-2t} u(-t)\\} = \\frac{-1}{s+2}$ with $ROC: \\text{Re}(s) < -2$"},{"type":"paragraph","content":"$\\mathcal{L}\\{e^{-3t} u(-t)\\} = \\frac{-1}{s+3}$ with $ROC: \\text{Re}(s) < -3$"},{"type":"paragraph","content":"The total transform $X(s)$ is the sum:"},{"type":"math","content":"X(s) = \\frac{-1}{s+2} - \\frac{1}{s+3} = \\frac{-(s+3) - (s+2)}{(s+2)(s+3)} = \\frac{-(2s+5)}{(s+2)(s+3)}"},{"type":"paragraph","content":"**Poles and Zeros:**\n- Poles (Denominator roots): $s = -2$ and $s = -3$.\n- Zero (Numerator root): $2s + 5 = 0 \\implies s = -2.5$."},{"type":"paragraph","content":"**Region of Convergence:** The overall ROC is the intersection of the two individual ROCs. Thus, the ROC is the region to the left of the leftmost pole in the s-plane: **$\\text{Re}(s) < -3$**."},{"type":"heading","content":"(b) Filtering"},{"type":"paragraph","content":"**Filtering** is the process of selectively altering the amplitude and phase characteristics of a signal's frequency components. A filter is an LTI system designed to pass a desired band of frequencies unattenuated while completely rejecting or highly attenuating all other unwanted frequencies (like noise). Common types include Low-Pass Filters (LPF), High-Pass Filters (HPF), and Band-Pass Filters (BPF)."}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Find the inverse z-transform of $x(z) = \\frac{z^3 - 10z^2 - 4z + 4}{2z^2 + 2z - 4}$ with ROC $|z| < 1$"}],"answer":{"summary":"The inverse z-transform is found using partial fraction expansion on X(z)/z, evaluating to an anti-causal sequence since |z|<1.","explanation":[{"type":"paragraph","content":"Given $X(z) = \\frac{z^3 - 10z^2 - 4z + 4}{2z^2 + 2z - 4}$ with ROC $|z| < 1$."},{"type":"paragraph","content":"First, simplify the denominator: $2z^2 + 2z - 4 = 2(z^2 + z - 2) = 2(z+2)(z-1)$."},{"type":"paragraph","content":"Since the degree of the numerator (3) is higher than the denominator (2), we first perform long division to express it as a polynomial plus a proper rational function:"},{"type":"math","content":"\\frac{z^3 - 10z^2 - 4z + 4}{2z^2 + 2z - 4} = \\frac{1}{2}z - \\frac{11}{4} + \\frac{13.5z - 7}{2(z+2)(z-1)}"},{"type":"paragraph","content":"Now, apply partial fraction expansion to the rational part:"},{"type":"math","content":"\\frac{13.5z - 7}{2(z+2)(z-1)} = \\frac{A}{z+2} + \\frac{B}{z-1}"},{"type":"paragraph","content":"Solving for A and B yields $A = \\frac{17}{3}$ and $B = \\frac{13}{12}$."},{"type":"paragraph","content":"Rewriting $X(z)$ in standard forms that allow easy lookup, we factor out $z^{-1}$ from the fractions so we can use the form $\\frac{z}{z-p}$:"},{"type":"math","content":"X(z) = \\frac{1}{2}z - \\frac{11}{4} + z^{-1} \\left[ \\frac{\\frac{17}{3}z}{z+2} + \\frac{\\frac{13}{12}z}{z-1} \\right]"},{"type":"paragraph","content":"The ROC is $|z| < 1$. The poles are at $z = -2$ and $z = 1$. Since $|z| < 1$ is strictly interior to both poles, both resulting time-domain sequences must be left-sided (anti-causal). Recall that for $|z| < |p|$, the inverse transform of $\\frac{z}{z-p}$ is $-p^n u[-n-1]$."},{"type":"paragraph","content":"Taking the inverse Z-transform term by term:\n1. $\\frac{1}{2}z \\implies \\frac{1}{2} \\delta[n+1]$\n2. $-\\frac{11}{4} \\implies -\\frac{11}{4} \\delta[n]$\n3. $z^{-1}$ introduces a time delay of 1 sample ($n \\to n-1$).\n4. Term for $z=-2$: $-\\frac{17}{3} (-2)^{n-1} u[-(n-1)-1] = -\\frac{17}{3} (-2)^{n-1} u[-n]$\n5. Term for $z=1$: $-\\frac{13}{12} (1)^{n-1} u[-(n-1)-1] = -\\frac{13}{12} u[-n]$"},{"type":"paragraph","content":"Combining the terms, the final sequence is:"},{"type":"math","content":"x[n] = \\frac{1}{2} \\delta[n+1] - \\frac{11}{4} \\delta[n] - \\frac{17}{3} (-2)^{n-1} u[-n] - \\frac{13}{12} u[-n]"}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"A signal x[n] with Fourier transform $x(e^{j\\omega})$ has the property that:\n\n$$\\left( x[n] \\sum_{k=-\\infty}^{\\infty} \\delta[n-3k] \\right) * \\left( \\frac{\\sin \\frac{\\pi}{3}n}{\\frac{\\pi}{3}n} \\right) = x[n]$$\n\nFor what values of w is it guaranteed that $x(e^{j\\omega}) = 0$."}],"answer":{"summary":"The equation represents sampling followed by ideal interpolation. Perfect reconstruction guarantees X(e^{jw}) must be strictly bandlimited to pi/3.","explanation":[{"type":"paragraph","content":"The given equation represents the process of discrete-time sampling and interpolation (reconstruction):"},{"type":"math","content":"\\left( x[n] \\sum_{k=-\\infty}^{\\infty} \\delta[n-3k] \\right) * \\left( \\frac{\\sin \\frac{\\pi}{3}n}{\\frac{\\pi}{3}n} \\right) = x[n]"},{"type":"paragraph","content":"1. The term $x[n] \\sum \\delta[n-3k]$ represents sampling the discrete-time signal $x[n]$ every $N = 3$ samples. This process creates periodic replicas of the original frequency spectrum $X(e^{j\\omega})$ shifted by integer multiples of $2\\pi/3$."},{"type":"paragraph","content":"2. The sampled signal is then convolved with an ideal discrete-time low-pass filter (represented by the sinc function) having a cutoff frequency of $\\omega_c = \\pi/3$."},{"type":"paragraph","content":"3. The equation states that the output of this low-pass filter exactly equals the original sequence $x[n]$."},{"type":"paragraph","content":"According to the **Nyquist Sampling Theorem**, for perfect reconstruction from sampled data without aliasing distortion, the original signal's frequency spectrum must not overlap when replicated. Therefore, the signal must be strictly band-limited to half the sampling frequency."},{"type":"paragraph","content":"Since the sampling period here is $N = 3$, the sampling frequency is $2\\pi/3$. The Nyquist limit is half of this, which is $\\pi/3$. To avoid aliasing and ensure perfect reconstruction via the ideal low-pass filter, the original spectrum $X(e^{j\\omega})$ must contain no frequency components above $\\pi/3$."},{"type":"paragraph","content":"Thus, it is guaranteed that **$X(e^{j\\omega}) = 0$ for $\\frac{\\pi}{3} \\le |\\omega| \\le \\pi$.**"}]}},{"id":"rtu-ee-ec-eee-s4-signals-and-systems-2024-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"For a continuous time periodic signal:\n\n$$x(t) = 2 + \\cos\\left[ \\frac{2\\pi}{3} \\right] t + 4\\sin\\left[ \\frac{5\\pi t}{3} \\right]$$\n\nDetermine the fundamental frequency $w_0$ and the Fourier series coefficient $a_k$ such that\n\n$$x[t] = \\sum_{k=-\\infty}^{\\infty} a_k \\cdot e^{j k w_0 t}$$"}],"answer":{"summary":"The fundamental frequency is pi/3. The non-zero Fourier coefficients are a_0=2, a_2=a_{-2}=0.5, and a_5=-2j, a_{-5}=2j.","explanation":[{"type":"paragraph","content":"Given the continuous-time periodic signal:"},{"type":"math","content":"x(t) = 2 + \\cos\\left( \\frac{2\\pi}{3} t \\right) + 4\\sin\\left( \\frac{5\\pi}{3} t \\right)"},{"type":"paragraph","content":"**Step 1: Determine the fundamental frequency $\\omega_0$.**"},{"type":"paragraph","content":"The signal consists of three components with frequencies $\\omega_1 = 0$, $\\omega_2 = 2\\pi/3$, and $\\omega_3 = 5\\pi/3$. The fundamental frequency $\\omega_0$ of the combined periodic signal is the Greatest Common Divisor (GCD) of the non-zero frequencies."},{"type":"math","content":"\\omega_0 = \\text{GCD}\\left( \\frac{2\\pi}{3}, \\frac{5\\pi}{3} \\right) = \\frac{\\pi}{3} \\text{ rad/s}"},{"type":"paragraph","content":"**Step 2: Express the signal in terms of complex exponentials.**"},{"type":"paragraph","content":"Using Euler's formulas: $\\cos(\\theta) = \\frac{e^{j\\theta} + e^{-j\\theta}}{2}$ and $\\sin(\\theta) = \\frac{e^{j\\theta} - e^{-j\\theta}}{2j}$"},{"type":"math","content":"x(t) = 2 + \\frac{1}{2} e^{j\\frac{2\\pi}{3}t} + \\frac{1}{2} e^{-j\\frac{2\\pi}{3}t} + \\frac{4}{2j} e^{j\\frac{5\\pi}{3}t} - \\frac{4}{2j} e^{-j\\frac{5\\pi}{3}t}"},{"type":"paragraph","content":"Rewrite the powers in terms of the fundamental frequency $\\omega_0 = \\pi/3$. Note that $\\frac{2\\pi}{3} = 2\\omega_0$ and $\\frac{5\\pi}{3} = 5\\omega_0$. Also, $1/j = -j$."},{"type":"math","content":"x(t) = 2 + \\frac{1}{2} e^{j(2)\\omega_0 t} + \\frac{1}{2} e^{-j(2)\\omega_0 t} - j2 e^{j(5)\\omega_0 t} + j2 e^{-j(5)\\omega_0 t}"},{"type":"paragraph","content":"**Step 3: Extract the Fourier Series Coefficients $a_k$.**"},{"type":"paragraph","content":"Comparing the expansion to the standard Fourier series equation $x(t) = \\sum_{k=-\\infty}^{\\infty} a_k e^{j k \\omega_0 t}$, we match the coefficients for each harmonic $k$:"},{"type":"paragraph","content":"- For $k = 0$ (DC component): $a_0 = 2$\n- For $k = 2$: $a_2 = 1/2$\n- For $k = -2$: $a_{-2} = 1/2$\n- For $k = 5$: $a_5 = -j2$\n- For $k = -5$: $a_{-5} = j2$"},{"type":"paragraph","content":"All other coefficients $a_k = 0$ for $k \\notin \\{0, 2, -2, 5, -5\\}$."}]}}]}],"loading":false,"error":null}}}

Q3Signals and Systems

Question

Answer