4.02 Spectral Replications and Aliasing Dynamics

4.01 The Shannon-Nyquist Sampling Theorem | 4.03 Ideal, Natural & Flat-Top Sampling Techniques

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4.02 Spectral Replications & Aliasing Dynamics

Core Idea

Uniform time-domain sampling mathematically translates to a periodic replication of the signal’s continuous spectrum in the frequency domain at integer multiples of the sampling frequency ${\Omega }_s$. The relationship between the sampling rate ${\Omega }_s$ and the maximum signal frequency ${\Omega }_m$ determines whether these replicated spectral bands remain isolated or overlap. When under-sampled, overlapping bands cause Aliasing, a destructive, irreversible frequency-folding distortion that can only be prevented prior to sampling using an analog Anti-Aliasing Filter.

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1. The Physics of Spectral Replications

To understand what happens in the frequency domain when we sample an analog signal, we recall the continuous-time impulse-train sampled signal $x_s(t)$ from Note 4.01: $x_s(t)=x(t)\cdot p(t)={\sum }_{n=-\infty }^{\infty }x(n{T}_s)\delta (t-n{T}_s)$

Since the sampling function $p(t)=\sum \delta (t-n{T}_s)$ is periodic with period $T_s$, its Fourier transform $P(j\Omega )$ is a discrete impulse train in the frequency domain spaced by the angular sampling frequency ${\Omega }_s=\frac{2\pi }{T_s}$: $P(j\Omega )=\frac{2\pi }{T_s}{\sum }_{k=-\infty }^{\infty }\delta (\Omega -k{\Omega }_s)$

Applying the frequency convolution property {multiplication in time is convolution in frequency}, we derive the spectrum of the sampled signal $X_s(j\Omega )$: $X_s(j\Omega )=\mathcal{F}\{x(t)\cdot p(t)\}=\frac{1}{2\pi }\left[X(j\Omega )\ast P(j\Omega )\right]$

$X_s(j\Omega )=\frac{1}{2\pi }\left[X(j\Omega )\ast \left(\frac{2\pi }{T_s}{\sum }_{k=-\infty }^{\infty }\delta (\Omega -k{\Omega }_s)\right)\right]$

$X_s(j\Omega )=\frac{1}{T_s}{\sum }_{k=-\infty }^{\infty }X\left(j\left(\Omega -k{\Omega }_s\right)\right)$

This elegant formula reveals that the spectrum of a sampled signal consists of the original spectrum $X(j\Omega )$ scaled by $\frac{1}{T_s}$ and repeated infinitely at every integer multiple of the sampling frequency ${\Omega }_s$.

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2. The Three Sampling Conditions

The boundary condition that determines the separation of these replicated spectral bands is established by comparing the angular sampling frequency ${\Omega }_s$ with twice the maximum frequency component of the analog signal ${\Omega }_m$.

2.1 Over-Sampling (${\Omega }_s>2{\Omega }_m$)

When the sampling frequency is strictly greater than twice the highest frequency component, the replicated spectral copies are spaced far apart.

• Guard Band: A clear frequency gap {buffer band} exists between adjacent replicas, spanning the interval: $\Delta \Omega ={\Omega }_s-2{\Omega }_m>0$
• Reconstruction Filter: Because of this gap, a practical, low-cost analog low-pass reconstruction filter with a gradual transition band can easily isolate the baseband spectrum $X(j\Omega )$ without introducing phase or amplitude distortion.

[GRAPH: Replicated frequency spectra under the over-sampling condition (\Omega_s > 2\Omega_m). Plot shows a series of triangular-shaped spectral bands of height 1/T_s centered at \Omega = 0, \pm\Omega_s, \pm2\Omega_s. The baseband triangle is bounded by -\Omega_m and \Omega_m. Clear gaps of width \Omega_s - 2\Omega_m representing guard bands are highlighted between adjacent triangles - source: K. Deergha Rao Fig 7.14]

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2.2 Nyquist-Rate Sampling (${\Omega }_s=2{\Omega }_m$)

This is the boundary condition where the sampling rate is exactly equal to twice the maximum frequency of the signal.

• Edge-Touching: The adjacent spectral replicas touch exactly at their edges with no frequency gap: ${\Omega }_s-2{\Omega }_m=0⟹{\Omega }_s-{\Omega }_m={\Omega }_m$
• The “Brick-Wall” Necessity: To reconstruct the original signal perfectly, we require an ideal low-pass filter with an infinitely sharp, vertical cutoff at ${\Omega }_c={\Omega }_m$: $H_{LP}(j\Omega )=\{\begin{array}{ll}{T}_s,& |\Omega |\le {\Omega }_m\\ 0,& |\Omega |>{\Omega }_m\end{array}$ Physical Constraint: Such a filter is mathematically causal but physically non-realizable because its impulse response is a double-sided sinc function extending from $t=-\infty$ to $t=\infty$, requiring infinite time delay (anticipatory memory) to implement.

[GRAPH: Replicated frequency spectra under the Nyquist-rate sampling condition (\Omega_s = 2\Omega_m). Plot shows triangular spectral bands of height 1/T_s centered at 0 and \pm\Omega_s touching exactly at the boundary frequency \Omega = \Omega_m, showing zero gap between the adjacent bands - source: K. Deergha Rao Fig 7.12]

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2.3 Under-Sampling (${\Omega }_s<2{\Omega }_m$)

When the sampling frequency is less than twice the maximum frequency component, the adjacent spectral replicas overlap in the frequency domain.

• Fold-Over Region: The high-frequency tails of one replica overlap with the low-frequency portion of the adjacent replica.
• Destructive Interference: Within this overlapping band, the spectra sum together algebraically. The original high frequencies fold back into the lower frequencies, making it physically impossible to reconstruct the original waveform. This spectral overlap region is called the fold-over region or aliasing zone.

[GRAPH: Replicated frequency spectra under the under-sampling condition (\Omega_s < 2\Omega_m) demonstrating aliasing. Plot shows overlapping triangular spectral bands centered at 0, \pm\Omega_s, \pm2\Omega_s. The overlapping regions between adjacent triangles are heavily shaded and labeled as 'Fold-over / Aliasing region' to show the corruption of low-frequency data by high-frequency replicas - source: K. Deergha Rao Fig 7.13]

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3. Aliasing Distortion & The Frequency-Folding Mechanism

3.1 Mathematical Definition of Aliasing

Aliasing is the frequency-folding distortion that occurs when a continuous-time signal is sampled at a rate below its Nyquist limit. High-frequency components “fold” around the folding frequency $f_{\text{fold}}=\frac{f_s}{2}$ {also known as the folding boundary or half-sampling limit} and reappear as false, lower-frequency components in the reconstructed baseband.

3.2 The Symmetrical Folding Formula

If a continuous-time sinusoidal signal has a true frequency $f_0$, and it is sampled at a rate $f_s$, the apparent reconstructed frequency $f_{\text{alias}}$ in the baseband $|f|\le \frac{f_s}{2}$ can be calculated using the symmetrical folding formula: $f_{\text{alias}}=\left|f_0-k{f}_s\right|\text{where }k\in \mathbb{Z}\text{ such that }{f}_{\text{alias}}\le \frac{f_s}{2}$

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3.3 High-Yield Algebraic Aliasing Proof

To see this folding mechanism algebraically, let us sample a high-frequency cosine wave $x(t)=\cos (2\pi f_{0}t)$ at a sampling rate $f_s=\frac{1}{T_s}$ where $f_s<2f_0$.

The discrete-time sequence of samples is: $x[n]=x(n{T}_s)=\cos \left(2\pi f_{0}n{T}_s\right)=\cos \left(2\pi \frac{f_0}{f_s}n\right)$

Let the true frequency be represented as an integer multiple of the sampling frequency plus an offset: $f_0=k{f}_s\pm f_{\text{alias}}⟹\frac{f_0}{f_s}=k\pm \frac{f_{\text{alias}}}{f_s}$

Substitute this ratio back into our discrete-time expression: $x[n]=\cos \left(2\pi \left(k\pm \frac{f_{\text{alias}}}{f_s}\right)n\right)$ $x[n]=\cos \left(2\pi kn\pm 2\pi \frac{f_{\text{alias}}}{f_s}n\right)$

Using the cosine angle addition identity $\cos (\theta \pm 2\pi kn)=\cos (\theta )$ for any integer $k$ and $n$: $x[n]=\cos \left(2\pi \frac{f_{\text{alias}}}{f_s}n\right)=\cos \left(2\pi f_{\text{alias}}n{T}_s\right)$

This algebraic identity proves that the discrete sample values of the high-frequency sinusoid $f_0$ are mathematically identical to the sample values of a lower-frequency sinusoid $f_{\text{alias}}$. During reconstruction, the DAC cannot distinguish between these two signals and will always output the lower frequency $f_{\text{alias}}$, resulting in irreversible data loss.

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3.4 Grounded Numerical Example

Problem: A continuous-time signal $x(t)=\cos (200\pi t)$ is sampled at a rate $f_s=150\text{ Hz}$. Find the true frequency of the signal, verify if aliasing occurs, and determine the apparent frequency of the reconstructed signal.

Step 1: Extract the True Frequency

From the expression $x(t)=\cos (2\pi f_{0}t)=\cos (200\pi t)$: $2\pi f_0=200\pi ⟹f_0=100\text{ Hz}$

Step 2: Test the Nyquist Criteria

The minimum theoretical sampling rate to avoid aliasing is the Nyquist Rate: $f_{\text{Nyquist}}=2f_0=2(100)=200\text{ Hz}$

Since our actual sampling frequency $f_s=150\text{ Hz}<200\text{ Hz}$, aliasing will definitely occur.

Step 3: Apply the Folding Formula

We seek an integer $k$ such that the aliased frequency falls below the half-sampling folding limit $f_{\text{fold}}=\frac{150}{2}=75\text{ Hz}$: For $k=1$: $f_{\text{alias}}=|f_0-f_s|=|100-150|=50\text{ Hz}$

Since $50\text{ Hz}\le 75\text{ Hz}$, the apparent reconstructed frequency is: $f_{\text{alias}}=50\text{ Hz}$

Step 4: Verify Graphically and Algebraically

$x[n]=\cos \left(2\pi \frac{100}{150}n\right)=\cos \left(\frac{4\pi }{3}n\right)$ $\cos \left(\frac{4\pi }{3}n\right)=\cos \left(2\pi n-\frac{2\pi }{3}n\right)=\cos \left(-\frac{2\pi }{3}n\right)=\cos \left(\frac{2\pi }{3}n\right)$ $\cos \left(\frac{2\pi }{3}n\right)=\cos \left(2\pi \frac{50}{150}n\right)=\cos \left(2\pi f_{\text{alias}}n{T}_s\right)$

The $100\text{ Hz}$ wave has folded around the $75\text{ Hz}$ folding frequency and is reconstructed as a false $50\text{ Hz}$ sine wave.

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4. The Anti-Aliasing Filter (AAF)

4.1 The Fundamental Post-Sampling Rule

The Post-Sampling Filtering Fallacy

Once a signal is sampled and aliasing occurs, the overlapping spectral components merge completely. At this point, the original low frequencies and folded-back high frequencies are mathematically summed into a single amplitude value at each frequency index. No digital filter or software algorithm can ever separate them. Aliasing is an irreversible physical process.

4.2 The Solution: Analog Pre-Filtering

To prevent aliasing, we must remove all frequency components above the half-sampling limit before the signal reaches the sampler or Analog-to-Digital Converter (ADC). This is achieved using an analog Anti-Aliasing Filter (AAF).

                 +-----------------------+      +---------+
Analog Input --->| Anti-Aliasing Filter  |----->| Sampler |---> Sampled Output
   x(t)          |  (Analog Low-Pass)    |      |  (ADC)  |        x[n]
                 +-----------------------+      +---------+
                         Cutoff:
                    f_c <= f_s / 2

4.3 Structural Design Specifications of an AAF

1. Filter Classification: Active or passive analog low-pass filter (LPF).
2. Cutoff Frequency ($f_c$): Designed to match the folding frequency boundary: $f_c\le \frac{f_s}{2}$
3. The Over-Sampling Relaxer: In practical engineering, designing a sharp analog filter with a very narrow transition band is incredibly difficult and expensive. Therefore, we deliberately over-sample the signal ($f_s\gg 2f_c$) to create a wide guard band. This wide guard band allows us to use a cheap, low-order analog anti-aliasing filter while still completely preventing aliasing.

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5. Common Mistakes That Cost Marks

The Post-ADC Digital Filtering Trap

In exam papers, examiners often ask if a digital filter implemented in a DSP can eliminate aliasing. Always answer NO. Explain that once the analog signal passes through the ADC sampler, the overlapping spectral copies are already summed. A digital filter cannot distinguish between a true $50\text{ Hz}$ signal and a $100\text{ Hz}$ signal aliased to $50\text{ Hz}$.

Incorrect Folding Calculations

When calculating folding frequencies, do not just subtract the frequency as $f_{\text{alias}}=f_s-f_0$. Always take the absolute value $|f_0-k{f}_s|$. Forgetting the absolute value yields a negative frequency, which is mathematically correct under exponential notation but physically incorrect for real-valued signals without a corresponding phase-sign inversion.

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6. PYQ Bank — Verbatim Questions & Answer Plans

6.1 PYQ 2021 Question 1c / 2019 Question 2b [06 Marks]

Question: Explain the aliasing effect and describe the means/filters used to avoid it.

• Answer Plan:
  1. Define Aliasing mathematically and conceptually (Section 3.1).
  2. State the condition for under-sampling ($f_s<2f_{\text{max}}$) which leads to spectral overlap (Section 2.3).
  3. Draw the overlapping triangular spectrum diagram (Section 2.3).
  4. Explain the solution: Placing an analog Anti-Aliasing Filter before the sampler with $f_c\le f_s/2$ (Section 4.2 & 4.3).

6.2 PYQ 2018 Question 2b [06 Marks]

Question: Draw the criteria and frequency spectrum in case of over-sampling, under-sampling & Nyquist rate sampling.

• Answer Plan:
  1. Divide your answer into three distinct sub-sections matching Section 2.
  2. For Over-sampling, state ${\Omega }_s>2{\Omega }_m$ and draw the spaced-out triangular replicas with highlighted guard bands.
  3. For Nyquist-rate sampling, state ${\Omega }_s=2{\Omega }_m$ and draw the replicas touching exactly.
  4. For Under-sampling, state ${\Omega }_s<2{\Omega }_m$ and draw overlapping replicas with shaded fold-over aliasing regions.

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7. Self-Check Before Moving On

• Can you derive the periodic replicated spectrum equation $X_s(j\Omega )$ using time-convolution?
• Do you know the exact frequency criteria for over-sampling, under-sampling, and Nyquist rate sampling?
• Can you calculate the apparent folding frequency of any under-sampled sinusoid using $f_{\text{alias}}=|f_0-k{f}_s|$?
• Can you explain why a digital filter cannot remove aliasing after a signal has been sampled?
• Do you understand the role, physical placement, and cutoff constraints of an analog Anti-Aliasing Filter?

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 7.4), ECE 2107 - Signals and Systems.md, 1.05 The Sampling Theorem Bridge & Multi-Rate Processing.md, Azmat Sir Class Slides.