05

Here is Note 5: The Sampling Theorem Bridge & Multi-Rate Processing.

Focus: This note is your critical bridge from continuous-time (analog) to discrete-time (digital) signals. Examiners heavily target this section because you cannot process digital signals without understanding how to sample them first. This note covers the mandatory theorems, the aliasing phenomenon, and the highly-tested up/down-sampling numericals.

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Topic 1: The Shannon-Nyquist Sampling Theorem

1.1 The Core Theorem

To convert an analog signal into a digital signal without losing any information, you must sample it fast enough.

• Statement: An analog signal must be sampled at a rate at least twice as large as the highest frequency component present in the analog signal to allow for perfect reconstruction.
• Mathematical Condition: $F_s\ge 2f_{max}$ where $F_s$ is the sampling frequency (or $F_T$) and $f_{max}$ (or $f_H$) is the maximum frequency of the signal.
• Nyquist Rate: The absolute minimum boundary condition for perfect reconstruction: $F_s=2f_{max}$.

1.2 The Three Sampling Conditions

Examiners frequently ask you to show the criteria and frequency spectrum for these three states. Do not write lengthy paragraphs; draw the spectrums.

1. Over-sampling ($F_s>2f_{max}$): The replicated frequency spectrums are spaced far apart. A simple, cheap low-pass filter can perfectly reconstruct the original signal.
2. Nyquist-rate sampling ($F_s=2f_{max}$): The replicated spectrums are exactly touching. An ideal (brick-wall) low-pass filter is required for reconstruction.
3. Under-sampling ($F_s<2f_{max}$): The replicated spectrums overlap and crash into each other. This causes Aliasing.

[Graph: Three separate frequency spectrum axes (\Omega). 1. Over-sampling showing triangles spaced apart. 2. Nyquist sampling showing triangles exactly touching at their bases. 3. Under-sampling showing the bases of the triangles overlapping (shaded region representing aliasing distortion)]

1.3 Aliasing Effect & Anti-Aliasing Filter

• What is Aliasing? A destructive frequency-folding distortion that occurs when a signal is under-sampled ($F_s<2f_{max}$). High-frequency components fold back and masquerade as false low-frequency components, permanently corrupting the signal.
• How to avoid it: Place an Anti-aliasing Filter (a strict analog low-pass filter) before the Analog-to-Digital Converter (ADC). This filter strips away any frequencies higher than $F_s/2$ before sampling even begins, ensuring the Nyquist criteria is never violated.

🔥 Exact PYQs for Topic 1:

• What is sampling theorem? (2024, 2023, 2022, 2016, 2015)
• State and prove the Shannon-Nyquist sampling theorem. [10/08 marks] (2019, 2018, 2017) (Note: You must memorize the formal mathematical proof of $x(t)\cdot {\delta }_T(t)$ mapping to convolution in the frequency domain).
• Explain aliasing effect and the means to avoid it. (2021, 2019, 2018, 2016, 2015)
• Write short notes on: (i) Over-sampling, (ii) Nyquist rate sampling, (iii) Aliasing. (2018, 2016)

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Topic 2: Classification of Sampling Techniques

When the examiner asks to classify sampling techniques, they are looking for the practical physical implementation of the sampling switch.

1. Ideal Sampling (Impulse Sampling): The analog signal is multiplied by a train of perfect, zero-width Dirac delta impulses. Physically impossible to build, used only for theoretical math.
2. Natural Sampling: The analog signal is multiplied by a train of rectangular pulses. The top of each sampled pulse naturally curves to follow the exact shape of the analog wave.
3. Flat-top Sampling: The analog signal is sampled and its voltage level is held strictly constant for the duration of the pulse width (Sample-and-Hold circuit). This creates a staircase-like waveform. It is the most practical method used in modern ADCs.

🔥 Exact PYQs for Topic 2:

• Classify the sampling techniques. (2018)
• Distinguish between Ideal, Natural, and Flat-top sampling techniques. (2017)

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Topic 3: Multi-Rate Signal Processing (Up/Down Sampling)

Examiners love this topic because students frequently mix up which operation deletes data and which operation adds zeros.

3.1 Discrete-Time Down-sampler (Compressor)

Used to reduce the sampling rate of a signal.

• Definition: A down-sampler by an integer factor $M$ keeps every $M$-th sample of the input sequence and permanently removes the $(M-1)$ in-between samples.
• Mathematical Operation: $x_d(n)=x(nM)$
• Warning: This operation actively discards data. If the original sequence was not sufficiently over-sampled, down-sampling will cause digital aliasing.

3.2 Discrete-Time Up-sampler (Interpolator)

Used to increase the sampling rate of a signal.

• Definition: An up-sampler by an integer factor $L$ inserts exactly $(L-1)$ zero-valued samples between each original sample of the input sequence.
• Mathematical Operation: $x_e(n)=\{\begin{array}{llc}x(n/L),& n=0,\pm L,\pm 2L,\dots 0,& \text{otherwise}\end{array}$

3.3 PYQ Master-Class: Array Numericals

When given an array, identify the origin $n=0$ (usually underlined or marked with an arrow).

🔥 Exact PYQ: What is discrete time up-sampler and down-sampler? If $x[n]=[2,6,\underline{0},3,1,2,...]$, then find the compressed and expanded signal of $x[n]$ by down-sampler 3 and up-sampler 3. (2022)

Step-by-Step Solution:

1. Map the indices: $x[-2]=2$, $x[-1]=6$, $x[\underline{0}]=0$, $x=3$, $x=1$, $x=2$
2. Down-sampler by 3 ($M=3$): Equation: $x_d(n)=x(3n)$. We only evaluate at multiples of 3.
   • For $n=0$: $x_d(0)=x(3\cdot 0)=x(0)=\underline{0}$
   • For $n=1$: $x_d(1)=x(3\cdot 1)=x(3)=2$
   • For $n=-1$: $x_d(-1)=x(3\cdot -1)=x(-3)$ (Not given, assume 0)
   • Result: $x_d(n)=[\dots 0,\underline{0},2,\dots ]$
3. Up-sampler by 3 ($L=3$): Equation: Insert $L-1=2$ zeros between every single original element.
   • Original: $2,6,\underline{0},3,1,2$
   • Result: $x_e(n)=[2,0,0,6,0,0,\underline{0},0,0,3,0,0,1,0,0,2]$

🔥 Remaining Exact PYQs for Topic 3:

• Define (i) Discrete time up-sampler; (ii) Discrete time down-sampler. (2021)
• Find out the up-sampling sequence $x_u(n)$ and down-sampling sequence $x_d(n)$ from the input signal sequence $x(n)=\dots$ for the discrete time system. [3+7 marks] (2024) (Note: Ensure you meticulously count the origin $n=0$ position when parsing the 2024 array).