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# System prepended metadata

title: Digital Signal Processing / Week1
tags: [DSP]

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# Digital Signal Processing / [Part 1](https://homepage.ntu.edu.tw/~r11942048/)
2023/03/11 in NTU by Potter
[Pionex派網]((https://www.pionex.com/zh-TW/signUp?r=3uU7AXmq))
edit by ChatGPT

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重點:
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1. **離散時間信號定義**：形式為 \( x = \{x[n]\}, -\infty < n < \infty, n \) 為整數，可透過對連續信號取樣得到。
2. **基本序列操作**：包含信號縮放、加法、乘法，及延遲或平移。
3. **基本序列類型**：包括單位樣本序列（衝擊信號）、單位步階序列、正弦和指數序列。
4. **離散時間系統定義**：將輸入序列映射至輸出序列的轉換或運算。
5. **系統特性**：涉及無記憶性、線性、時不變性、因果性和穩定性。
6. **線性時不變（LTI）系統**：對脈衝響應的完全特徵描述，並藉由卷積和來表達系統輸出。
7. **穩定性的判斷**：系統穩定性與脈衝響應的絕對可加性有關。

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模擬試題:
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Multiple Choice Question:
1. When a discrete-time signal \( x[n] \) is obtained through sampling and the sampling frequency is insufficient to satisfy the Nyquist theorem, which phenomenon is likely to occur?
   A. Interpolation
   B. Oversampling
   C. Aliasing
   D. Spectrum Inversion

**Correct Answer: C. Aliasing**
**Explanation:** Aliasing is a phenomenon that occurs when a signal is sampled at a rate lower than twice its highest frequency (Nyquist rate), resulting in overlapping frequency spectra and the creation of incorrect frequencies in the sampled signal.

True/False Question:
2. In discrete-time systems, if the output is only related to the current input and independent of past or future inputs, the system is called a memoryless system. (True/False)

**Correct Answer: True**
**Explanation:** Memoryless systems respond solely based on the current input without any dependency on previous inputs or future inputs, which is indicative of the system having no memory or storage capability of past values.

Multiple Choice Question:
3. According to the definition of linear systems, which of the following operations does not satisfy the property of a linear system?
   A. Superposition Principle
   B. Homogeneity Principle
   C. Scaling Operation
   D. Exponential Operation

**Correct Answer: D. Exponential Operation**
**Explanation:** Linear systems adhere to the superposition and homogeneity principles. The exponential operation is nonlinear because it does not proportionally scale the output when the input is scaled; thus, it does not satisfy linearity.

True/False Question:
4. For a stable Linear Time-Invariant (LTI) system, the impulse response \( h[n] \) must satisfy absolute summability, meaning \( \sum_{n=-\infty}^{\infty} |h[n]| \) is finite. (True/False)

**Correct Answer: True**
**Explanation:** The stability of an LTI system is determined by the Bounded-Input Bounded-Output (BIBO) criterion. For the system to be stable, its impulse response must be absolutely summable to ensure that every bounded input produces a bounded output.

Multiple Choice Question:
5. If the system output \( y[n] \) is the convolution of an input \( x[n] \) with the system \( H \), what type of system is \( H \)?
   A. Nonlinear System
   B. Non-Time-Invariant System
   C. Linear Time-Invariant System
   D. Causal System

**Correct Answer: C. Linear Time-Invariant System**
**Explanation:** The convolution operation is a hallmark of Linear Time-Invariant (LTI) systems. If a system's output can be expressed as the convolution of the input with the system's impulse response, the system exhibits both linearity (because of superposition) and time invariance (because the response is independent of when the input is applied).

