Related Concepts: 1.01 Signals, Systems & Singularity Functions | 1.02 Elementary Operations on Signals | 1.03 Signal Classification I — Theory & Periodicity | 1.04 Signal Classification II — Symmetry, Energy & Power

1.05 The Sampling Theorem Bridge & Multi-Rate Processing

Overview

This note serves as the critical bridge from continuous-time (analog) to discrete-time (digital) signals. Understanding how to sample an analog waveform without data loss is a fundamental requirement of signal processing.

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1. The Shannon-Nyquist Sampling Theorem

1.1 The Core Theorem

To convert a continuous-time analog signal into a discrete-time digital signal without losing any information, the analog wave must be sampled at a sufficiently high rate.

Shannon-Nyquist Sampling Theorem

A continuous-time, band-limited signal containing no frequency components higher than $f_{max}$ Hz can be uniquely and perfectly reconstructed from its samples if it is sampled at a rate $F_s$ samples per second that is at least twice the maximum frequency component.

$F_s\ge 2f_{max}$

• Nyquist Rate: The absolute minimum theoretical sampling rate that avoids information loss: $F_N=2f_{max}$
• Nyquist Interval: The maximum allowable time duration between consecutive samples: $T_N=\frac{1}{F_N}=\frac{1}{2f_{max}}$

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1.2 The Three Sampling Conditions

The relation between the sampling frequency $F_s$ and the Nyquist rate $2f_{max}$ determines the frequency-domain reconstruction behavior.

1. Over-sampling ($F_s>2f_{max}$)

The replicated spectral copies of the signal are spaced far apart. A guard band exists between them, allowing reconstruction with a cheap, non-ideal low-pass filter.

          Replicated Spectrums (Spaced apart with Guard Bands):
          ^                       ^                       ^
         / \                     / \                     / \
        /   \                   /   \                   /   \
   ----+-----+----+-------+----+-----+----+-------+----+-----+----+--> Frequency (Ω)
     -2f_max      0    2f_max                     F_s
                  |<----->|
                   Guard Band

2. Nyquist-rate sampling ($F_s=2f_{max}$)

The replicated spectral copies are exactly touching at their edges. An ideal “brick-wall” low-pass filter is required to reconstruct the original signal.

          Nyquist-rate sampling (Spectrums exactly touching):
          ^                       ^                       ^
         / \                     / \                     / \
        /   \                   /   \                   /   \
   ----+-----+----+-----------+----+-----+----+-----------+----+-----+----+--> Frequency (Ω)
            -f_max 0     f_max               F_s
                   |<-------->|
                     F_s = 2f_max

3. Under-sampling ($F_s<2f_{max}$)

The replicated spectral copies overlap. This causes Aliasing, preventing accurate reconstruction of the original signal.

          Under-sampling (Spectrums overlapping):
          ^               ^               ^
         / \             / \             / \
        /   \XXXXX      /   \XXXXX      /   \
   ----+-----+----+----+-----+----+----+-----+----+--> Frequency (Ω)
             |XXXXXXXXX|
           Overlapping Region (Aliasing Distortion)

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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 begins, ensuring the Nyquist criteria is never violated.

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Past Year Questions (PYQs)

• [PYQ 2024, 2023, 2022, 2016, 2015]: What is sampling theorem? (04 Marks)
• [PYQ 2019, 2018, 2017]: State and prove the Shannon-Nyquist sampling theorem. (08 to 10 Marks)
  • Note: Ensure you memorize the formal mathematical proof of $x_s(t)=x(t)\cdot {\delta }_T(t)$ mapping to frequency-domain convolution.
• [PYQ 2021, 2019, 2018, 2016, 2015]: Explain the aliasing effect and the means to avoid it. (06 Marks)
• [PYQ 2018, 2016]: Write short notes on: (i) Over-sampling, (ii) Nyquist rate sampling, (iii) Aliasing. (06 Marks)

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2. Classification of Sampling Techniques

Physical implementation of the sampling switch is categorized into three types:

1. Ideal Sampling (Impulse Sampling): The analog signal is multiplied by a train of perfect, zero-width Dirac delta impulses. This is physically impossible to build and is used only for theoretical mathematical modeling.
2. Natural Sampling: The analog signal is multiplied by a train of finite-width rectangular pulses. The top of each sampled pulse naturally curves to follow the exact contour 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 (using a Sample-and-Hold circuit). This creates a staircase-like waveform and is the standard method in modern ADCs.

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Past Year Questions (PYQs)

• [PYQ 2018]: Classify the sampling techniques. (04 Marks)
• [PYQ 2017]: Distinguish between Ideal, Natural, and Flat-top sampling techniques. (06 Marks)

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

Multi-rate processing manipulates the sampling rate of a discrete-time sequence.

3.1 Discrete-Time Down-sampler (Compressor)

Used to reduce the sampling rate.

• Definition: A down-sampler by an integer factor $M$ keeps every $M$-th sample of the input sequence and discards the $(M-1)$ intermediate samples.
• Mathematical Operation: $x_d[n]=x[nM]$
• Warning: Because this operation discards data, down-sampling will cause digital aliasing unless the original signal is sufficiently over-sampled.

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3.2 Discrete-Time Up-sampler (Interpolator)

Used to increase the sampling rate.

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

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3.3 Array Numerical Worked Example

High-Yield Array Numerical

Question: If $x[n]=[2,6,\underline{0},3,1,2]$, find the down-sampled sequence by a factor of 3, and the up-sampled sequence by a factor of 3. (2022)

Solution:

1. Index Mapping: Identify the origin $n=0$ (underlined): $x[-2]=2,x[-1]=6,x[\underline{0}]=0,x[1]=3,x[2]=1,x[3]=2$

2. Down-sampling ($M=3$): Keep only indices that are multiples of 3 ($x_d[n]=x[3n]$):

   • $n=0⟹x_d[0]=x[0]=\underline{0}$
   • $n=1⟹x_d[1]=x[3]=2$
   • $n=-1⟹x_d[-1]=x[-3]$ (Out of bounds, assume 0)

   ${\mathbf{x}}_{\mathbf{d}}[\mathbf{n}]=[\underline{0},2]$

3. Up-sampling ($L=3$): Insert $L-1=2$ zeros between each element of the original sequence:

   ${\mathbf{x}}_{\mathbf{e}}[\mathbf{n}]=[2,0,0,6,0,0,\underline{0},0,0,3,0,0,1,0,0,2]$

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Past Year Questions (PYQs)

• [PYQ 2021]: Define: (i) Discrete time up-sampler; (ii) Discrete time down-sampler. (04 Marks)
• [PYQ 2024]: 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. (10 Marks)