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A linear discrete time system obeys the property of superposition. The superposition property is composed of homogeneity and additivity. Let

x 1 [ n ] excitation produce y 1 [ n ] response and x 2 [ n ] produce y 2 [ n ] response. According

to additivity property of superposition theorem, if both x 1 [ n ] and x 2 [ n ] are applied

simultaneously, then

x 1 [ n ] + x 2 [ n ] = y 1 [ n ] + y 2 [ n ]

Let a 1 x 1 [ n ] and a 2 x 2 [ n ] be the inputs. According the homogeneity (scaling) property,

when these signals are separately applied,

a 1 x 1 [ n ] = a 1 y 1 [ n ]

a 2 x 2 [ n ] = a 2 y 2 [ n ]

If a 1 x 1 [ n ] + a 2 x 2 [ n ] are simultaneously applied, the output is obtained by applying

superposition theorem as,

a 1 x 1 [ n ] + a 2 x 2 [ n ] = a 1 y 1 [ n ] + a 2 y 2 [ n ]

In the above equation, a 1 x 1 [ n ] + a 2 x 2 [ n ] is called the weighted sum of input, and

a 1 y 1 [ n ] + a 2 y 2 [ n ] is called the weighted sum of the output.

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1. Check your group for 12/19 and the assigned features of the system.

2. Search online how the assigned feature is for the system.

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A discrete time system is said to be stable if for any bounded input, it produces a bounded output. This implies that the impulse response

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A discrete time system is said to be static (memoryless or instantaneous) if the output response depends on the present value only and not on the past and future values of excitation.

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Discrete systems described by difference equations

require memory, and hence they are dynamic systems.

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1. Please sit with your group members for 12/19.

2. Please read through the assigned paragraph/information of the System.

3. Describe the assigned system in Chinese.

4. Explain (orally or in written) whether the following exmples show the assigned feature or not.

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A discrete time system is said to be causal if the response of the system depends on the present or the past inputs applied.

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A system is generally de ﬁ ned as any group of elements or components that operates on one or more

input signals to produce one or more output signals. The idea of systems and their analysis is

pervasive not only in engineering and the physical sciences, but also in other disciplines. For

example, we are familiar with the non-engineering ideas of a biological system, a social system and

an economic system. Implicit in all de ﬁ nitions of systems is the idea that the inputs (e.g., force,

voltage, body temperature, industrial capacity) can be measured and that the system acts on those

inputs to produce outputs (e.g., displacement, current, blood pressure, GDP) in a manner that can

be modeled and quanti ﬁ ed. In digital signal processing, a discrete-time system is a formula, an

algorithm, a mathematical operation or a piece of code that transforms an input sequence into

an output sequence.

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Schematic description of a transformation

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Block diagram and signal representation to illustrate time invariancy of DT system

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If an arbitrary excitation x [ n ] of a system causes a response y [ n ] and the delayed excitation x [ n − n0 ] where n0 is any arbitrary integer causes y [ n − n0 ], then the system is said to be time invariant.

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1. y [ n ] = x [ n − 1 ]

2. y [ n − 1 ] = x [ n ]

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1. y [ n ] = sin x [ n ]

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1. y [ n ] = x [ 3n ]

2. y [ n ] = sin ( x [ n ])

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1. y [ n − 1 ] + y [ n ] = x [ n ]

2. y [ n ] = sgn | x [ n ]|

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1. y [ n ] = x [− n ]

2. y [ n ] = sin ( x [ n ])

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The idea of System in Digital Signal Processing.

1. Definition of the System in Digital Signal Processing.

2. Different features of the system.

3. How the features affect the processed signal.

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當今天一個訊號X[n]我們帶入一個時間變數n

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當今天一個訊號X[n]我們帶入一個時間變數n可以得到一個Y[n]的結果，而如果今天這個訊號是有時間延遲的，則這輸入會變成X[n-n0]，那如果這個結果剛好會滿足Y[n-n0]則稱為 time invariant system (非時變系統)

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Upload the VIDEO CLIP to explain your solution

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Upload the VIDEO CLIP to explain your solution

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1. Please record your explanation

2. Upload the clip by 5 pm on Fri (12/22)

3. Type your student ID as the title on Padlet to get the speaker's points

4. For the ones who haven't earned the points for yourselves, please seize the LAST chance

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The presence of a mechanism in the system that retains or stores information about input values at times other than the current time.

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線性離散時間系統遵從著加法性質。加法性質由均

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線性離散時間系統遵從著加法性質。加法性質由均質性和可加性構成。假設 x1[n] 的激勵產生 y1[n] 的響應，x2[n] 產生 y2[n] 的響應。根據加法性質的超複合定理，如果同時應用 x1[n] 和 x2[n]，則有：

x1[n] + x2[n] = y1[n] + y2[n]

令 a1x1[n] 和 a2x2[n] 為輸入。根據均質性（縮放）性質，在分別應用這些信號時，有：

a1x1[n] = a1y1[n]

a2x2[n] = a2y2[n]

如果同時應用 a1x1[n] + a2x2[n]，則輸出可通過應用超複合定理獲得：

a1x1[n] + a2x2[n] = a1y1[n] + a2y2[n]

在上述方程中，a1x1[n] + a2x2[n] 被稱為輸入的加權和，a1y1[n] + a2y2[n] 被稱為輸出的加權和。

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靜態系統隨著時間的推移保持不變或處於穩定狀熊

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靜態系統隨著時間的推移保持不變或處於穩定狀熊。在靜態系統中，系統的巒

數或屬性不隨時間或任何其他自變數而變化或改變。靜態系統的特徵是平衡、穩定、不存在運動或變化。

y[n]=(x[3n]) 也代表一個非靜態系統。在這個系統中，輸出 y[n] 取決於輸入 x[3n]，表示輸出信號的取樣點是輸入信號每隔三個時間點的取樣。因此，系統的輸出不僅取決於當前的輸入，還取決於時間的總體結構，這顯示出非靜態性質。

y[n]=sin(x[n]) 有機會代表一個靜態系統。雖然因為輸出 y[n] 是根據當前時間點 n 的輸入值 x[n] 的正弦函數，不過因為他某段時間內波形是固定的，所以有機會。

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How good is ChatGPT for the translation in Chinese?

Could ChatGPT give you some examples?