Signals and Systems

7.3 Undersampling, Aliasing, and Digital Signal Processing Fundamentals

Imron Rosyadi

Undersampling, Aliasing, and Digital Signal Processing Fundamentals

Bridging Continuous and Discrete Worlds

Welcome to this lecture on critical concepts in Signals and Systems. Today, we’ll dive deep into sampling, specifically focusing on the effects of undersampling, known as aliasing, and how we leverage these principles for digital processing of both continuous and discrete-time signals. These topics are fundamental to almost every area of Electrical and Computer Engineering.

Introduction: The Importance of Sampling

• Connecting Worlds: Sampling is the bridge between continuous-time (analog) signals and discrete-time (digital) signals.
• Digital Advantage: Allows us to process analog signals using powerful and flexible digital systems (computers, microcontrollers, DSPs).
• Core Concept: How do we represent a continuous signal with a finite set of discrete values?
• Reconstruction: How do we get the original continuous signal back from its samples?
• The Sampling Theorem: The theoretical foundation for perfect reconstruction.

Figure 7.19: Discrete-time processing of continuous-time signals.

Sampling is a cornerstone of modern ECE. It’s what allows us to take a real-world analog signal, like your voice or a sensor reading, and convert it into a digital format that can be stored, processed, and transmitted by computers. The advantages of digital processing are immense: flexibility, noise immunity, and the ability to implement complex algorithms. But to do this correctly, we need to understand the fundamental principles, especially the sampling theorem, which tells us the conditions under which we can perfectly reconstruct the original signal.

Recap: The Sampling Theorem

• Condition for Perfect Reconstruction: A band-limited signal \(x(t)\) with highest frequency \(\omega_M\) can be perfectly reconstructed from its samples \(x(nT)\) if the sampling frequency \(\omega_s\) is strictly greater than twice the highest frequency. \[ \omega_s > 2\omega_M \]
• Nyquist Rate: The minimum sampling rate, \(2\omega_M\), is called the Nyquist rate.
• Ideal Reconstruction: Achieved by passing the sampled signal through an ideal lowpass filter with a cutoff frequency \(\omega_c\) such that \(\omega_M < \omega_c < \omega_s - \omega_M\).

Why “strictly greater than”?

As we’ll see, sampling exactly at \(2\omega_M\) can lead to issues, especially with phase.

The Nyquist-Shannon Sampling Theorem is perhaps one of the most important theorems in signal processing. It states that if your signal is band-limited, meaning it doesn’t contain any frequencies above a certain maximum, ωM, then you can perfectly reconstruct it from its samples, provided you sample at a rate ωs that is greater than 2ωM. This 2ωM is known as the Nyquist rate. The reconstruction process involves an ideal lowpass filter, which effectively “selects” the baseband component from the periodic spectrum of the sampled signal. The “strictly greater than” part is important, as we’ll see with aliasing, and even sampling exactly at 2ωM can cause issues.

Undersampling: The Effect of Aliasing

• Definition: Aliasing occurs when the sampling frequency \(\omega_s\) is less than twice the highest frequency \(\omega_M\) in the signal ($ _s < 2_M $).
• Consequence: The spectral replicas of the original signal \(X(j\omega)\) in the sampled signal’s spectrum \(X_p(j\omega)\) overlap.
  • This overlap means the original signal’s spectrum is no longer recoverable by lowpass filtering.
• Time Domain: While the reconstructed signal \(x_r(t)\) will still match the original signal \(x(t)\) at the sampling instants (\(x_r(nT) = x(nT)\)), it will not be equal to \(x(t)\) between samples.

Figure 7.15 (d) Spectrum of sampled signal with ωs < 2ω0: Spectral overlap due to undersampling.

Aliasing is generally undesirable!

It introduces distortion where higher frequencies in the original signal are indistinguishable from lower frequencies after sampling, making perfect reconstruction impossible.

Aliasing is the primary problem associated with undersampling. When your sampling rate is too low, the spectral copies of your signal in the frequency domain start to overlap. Imagine multiple overlapping patterns – you can’t cleanly separate them anymore. This overlap means that high-frequency components in the original signal effectively “fold over” into the lower frequency range, masquerading as different, lower frequencies. While the samples themselves might align, the reconstructed signal between samples will be incorrect. This is a critical concept to avoid in most practical applications.

Aliasing: Frequency Domain Perspective (Sinusoidal Input)

• Consider \(x(t) = \cos(\omega_0 t)\). Its spectrum \(X(j\omega)\) has impulses at \(\pm\omega_0\).
• When sampled, \(X_p(j\omega)\) contains replicas of \(X(j\omega)\) centered at \(k\omega_s\).
• No Aliasing: If \(\omega_0 < \omega_s/2\), the replicas don’t overlap. The lowpass filter correctly extracts \(\cos(\omega_0 t)\).
  • E.g., \(\omega_0 = \omega_s/6\) or \(\omega_0 = 2\omega_s/6\).
• Aliasing Occurs: If \(\omega_0 > \omega_s/2\), the replicas overlap.
  • The lowpass filter extracts a different frequency.
  • For \(\omega_s/2 < \omega_0 < \omega_s\), the reconstructed signal is \(x_r(t) = \cos((\omega_s - \omega_0)t)\).
  • The original frequency \(\omega_0\) is “aliased” to a lower frequency \(\omega_s - \omega_0\).

Note

This phenomenon is often called frequency folding, as frequencies above \(\omega_s/2\) are folded back into the baseband spectrum.

Let’s make this concrete with a simple sinusoidal signal. The Fourier transform of a cosine is just two impulses in the frequency domain. When we sample this, these impulses are replicated periodically. If the sampling frequency is high enough, these replicas are separated, and an ideal lowpass filter can perfectly reconstruct the original signal. However, if the signal frequency is too high relative to the sampling frequency (i.e., ω0 > ωs/2), these replicas overlap. The lowpass filter will then pick up a different, lower frequency component from the aliased spectrum, leading to a distorted reconstructed signal. This is the essence of frequency folding.

Aliasing: Time Domain Impact & Phase Reversal

• Time Domain Visualization: The lowpass filter, when aliasing occurs, attempts to fit a sinusoid of frequency less than \(\omega_s/2\) to the sampled points.
  • This results in a reconstructed signal that has a different frequency than the original.
• Phase Reversal: For \(x(t) = \cos(\omega_0 t + \phi)\), aliasing can also cause a phase reversal.
  • When \(\omega_s/2 < \omega_0 < \omega_s\), the reconstructed signal might be \(x_r(t) = \cos((\omega_s - \omega_0)t - \phi)\).
  • The sign of the phase \(\phi\) is reversed.

Figure 7.16 (c) Aliasing with ω0 = 4ωs/6: Original signal (solid), samples, and reconstructed signal (dashed) showing aliasing.

The Nyquist-Shannon Theorem requirement is strictly \(\omega_s > 2\omega_M\).

Sampling exactly at \(2\omega_M\) is not sufficient.

Example 7.1:

• \(x(t) = \cos((\omega_s/2)t + \phi)\) sampled at \(\omega_s\).
• Reconstructed: \(x_r(t) = (\cos \phi) \cos((\omega_s/2)t)\).
• Perfect reconstruction only if \(\phi = 0\).
• If \(x(t) = \sin((\omega_s/2)t)\) (\(\phi = -\pi/2\)), then \(x_r(t) = 0\). Samples are all zero!

The time-domain effect of aliasing is visually striking. The lowpass filter, which is supposed to reconstruct the original signal, instead produces a completely different one, one that simply connects the dots of the samples with a lower frequency sinusoid. This can be misleading. Furthermore, if the signal has a phase component, aliasing can even reverse its sign, adding another layer of distortion. Example 7.1 highlights why the “strictly greater than” condition in the sampling theorem is crucial. If you sample a sine wave at exactly twice its frequency, you might get all zero samples, leading to zero reconstruction! This is a classic trap in sampling.

Interactive Demo: Understanding Aliasing

Adjust the signal frequency and sampling frequency to observe aliasing.

viewof sig_freq = Inputs.range([0.1, 5.0], {value: 1, step: 0.1, label: "Signal Frequency (Hz)"});
viewof samp_freq = Inputs.range([2.0, 20.0], {value: 5, step: 0.5, label: "Sampling Frequency (Hz)"});

This interactive demo allows you to directly manipulate the signal and sampling frequencies. Observe how the “Reconstructed Signal” (green dashed line) behaves. When the sampling frequency is sufficiently high (more than twice the signal frequency), it closely matches the original blue signal. However, as you decrease the sampling frequency below the Nyquist rate, you’ll clearly see the green dashed line representing a different, lower frequency than the original. This is aliasing in action. Try setting the signal frequency to 4 Hz and the sampling frequency to 5 Hz. The original 4 Hz signal will appear as a 1 Hz signal in the reconstruction (5 - 4 = 1).

Real-World Example: The Stroboscopic Effect

• Analogy: A strobe light illuminating a rotating object, or camera frames capturing a moving object.
• The strobe/camera acts as a sampling system, capturing brief snapshots at a periodic rate.
• Perception of Aliasing:
  • If strobe frequency \(\omega_s\) is much higher than rotational speed \(\omega_{rot}\), motion is perceived correctly.
  • If \(\omega_s < 2\omega_{rot}\), the rotation appears slower, or even in the wrong direction (due to phase reversal)!
• Examples:
  • Car wheels in old Western movies appearing to spin backward.
  • Helicopter blades appearing stationary or slow-moving in videos.

The stroboscopic effect is a fascinating real-world example of aliasing. Think about watching old movies where wagon wheels appear to spin backward. This isn’t magic; it’s aliasing. The camera captures discrete frames at a certain rate (the sampling frequency). If the wheel’s rotation speed (the signal frequency) is too high relative to the frame rate, the wheel’s spokes will appear to move backward. This is because the sampling rate is too low, and the higher rotational frequency is aliased to a lower, negative frequency, which we perceive as backward motion. It’s a vivid demonstration of how undersampling can completely alter our perception of motion.

Discrete-Time Processing of Continuous-Time Signals

• Motivation: Leverage digital systems (DSPs, microprocessors) for flexible and powerful signal processing.
• Overall System: A cascade of three operations:
  1. C/D (Continuous-to-Discrete) Converter: Transforms \(x_c(t)\) to \(x_d[n]\).
  2. Discrete-Time System: Processes \(x_d[n]\) to \(y_d[n]\) (e.g., digital filter).
  3. D/C (Discrete-to-Continuous) Converter: Transforms \(y_d[n]\) back to \(y_c(t)\).

This diagram illustrates the typical workflow for processing continuous-time signals using digital methods. The analog signal first goes through a C/D converter, often an Analog-to-Digital (A/D) converter, which samples and quantizes it into a discrete-time sequence. This sequence is then fed into a discrete-time system, which could be a digital filter, an equalizer, or a complex algorithm running on a DSP chip. Finally, the processed discrete-time signal is converted back into an analog signal using a D/C converter, or Digital-to-Analog converter, followed by a reconstruction filter. This entire process relies heavily on the principles of sampling we’ve discussed.

C/D and D/C Conversion Details

C/D Conversion:

1. Periodic Sampling: \(x_c(t)\) is multiplied by an impulse train \(p(t) = \sum \delta(t-nT)\) to get \(x_p(t)\).
2. Impulse-to-Sequence Mapping: The impulse train \(x_p(t)\) is mapped to a discrete-time sequence \(x_d[n]\) where \(x_d[n] = x_c(nT)\).
   • This effectively normalizes the time axis.

D/C Conversion:

1. Sequence-to-Impulse Mapping: \(y_d[n]\) is converted into an impulse train \(y_p(t) = \sum y_d[n]\delta(t-nT)\).
2. Lowpass Filtering: The impulse train \(y_p(t)\) is passed through an ideal lowpass filter to reconstruct \(y_c(t)\).
   • This interpolates between the sample values.

Let’s break down the C/D and D/C processes. The C/D conversion starts with periodic sampling, effectively taking snapshots of the continuous signal. These snapshots are then represented as a discrete sequence. The key idea here is normalizing the time axis: the continuous time t becomes discrete index n, and the sampling period T becomes the unit step.

The D/C conversion reverses this. The discrete sequence is first converted back into an impulse train, where each impulse’s amplitude is the sample value. Then, this impulse train is smoothed out by an ideal lowpass filter, which performs band-limited interpolation, reconstructing the continuous signal. This is where the sampling theorem’s conditions become critical for perfect reconstruction.

Frequency Domain Perspective of C/D Conversion

• Spectrum of Sampled Impulse Train (\(X_p(j\omega)\)): \[ X_p(j\omega) = \frac{1}{T} \sum_{k=-\infty}^{+\infty} X_c\left(j(\omega - k\omega_s)\right) \]
  • This shows \(X_p(j\omega)\) as periodic replicas of \(X_c(j\omega)\).
• Spectrum of Discrete-Time Sequence (\(X_d(e^{j\Omega})\)): \[ X_d(e^{j\Omega}) = \sum_{n=-\infty}^{+\infty} x_c(nT) e^{-j\Omega n} \]
• Relationship between \(X_p(j\omega)\) and \(X_d(e^{j\Omega})\): \[ X_d(e^{j\Omega}) = X_p(j\Omega/T) \]
  • The discrete-time frequency \(\Omega\) is related to the continuous-time frequency \(\omega\) by \(\Omega = \omega T\).
  • This means \(X_d(e^{j\Omega})\) is a frequency-scaled version of \(X_p(j\omega)\).

Tip

The scaling \(\Omega = \omega T\) is crucial for understanding how continuous-time frequencies map to discrete-time frequencies. The range \([-\omega_s/2, \omega_s/2]\) in continuous time maps to \([-\pi, \pi]\) in discrete time.

Understanding the frequency domain transformation is key. When we sample, the continuous-time spectrum Xc(jω) becomes replicated periodically in Xp(jω). Then, when we convert this impulse train into a discrete-time sequence, the spectrum Xp(jω) undergoes a frequency scaling to become Xd(e^jΩ). This scaling maps the continuous-time frequency axis to the discrete-time frequency axis. For instance, the Nyquist frequency ωs/2 in continuous time maps to π in discrete time. This relationship is fundamental for designing discrete-time filters that mimic continuous-time behavior.

Equivalent Continuous-Time System

• When the input \(x_c(t)\) is band-limited and the sampling theorem is satisfied, the overall system (C/D -> Discrete-Time System -> D/C) behaves like a continuous-time LTI system.
• Its equivalent continuous-time frequency response \(H_c(j\omega)\) is related to the discrete-time system’s frequency response \(H_d(e^{j\Omega})\) by: \[ H_c(j\omega) = \begin{cases} H_d(e^{j\omega T}), & |\omega| < \omega_s/2 \\ 0, & |\omega| > \omega_s/2 \end{cases} \]
• This relationship is the foundation for implementing continuous-time filters using discrete-time filters.

Equivalent Continuous-Time System

Figure 7.25(f) Yc(jω) spectrum

Figure 7.26 Hc(jω) vs Hd(e^jΩ): Relationship between discrete-time and equivalent continuous-time frequency responses.

This is a powerful result! It means that even though our system involves non-linear and time-varying operations like sampling, for band-limited inputs and appropriate sampling rates, the entire cascade can be modeled as a simple continuous-time LTI system. The key is that the discrete-time filter’s frequency response Hd(e^jΩ) gets scaled and effectively “copied” into the baseband of the continuous-time frequency axis to form Hc(jω). This allows ECE engineers to design complex continuous-time filters by designing their discrete-time counterparts, which are often easier to implement digitally.

Application: Digital Differentiator

• Goal: Implement a system where \(y_c(t) = \frac{d}{dt} x_c(t)\).
• Continuous-Time Ideal Differentiator: \(H_c(j\omega) = j\omega\).
• Band-Limited Differentiator: \[ H_c(j\omega) = \begin{cases} j\omega, & |\omega| < \omega_c \\ 0, & |\omega| > \omega_c \end{cases} \]
• Corresponding Discrete-Time Filter: Using \(\Omega = \omega T\) and \(\omega_s = 2\omega_c\), the discrete-time frequency response is: \[ H_d(e^{j\Omega}) = j\left(\frac{\Omega}{T}\right), \quad |\Omega| < \pi \]

Application: Digital Differentiator

Figure 7.27 Continuous-time ideal band-limited differentiator

Ideal \(H_c(j\omega)\)

Figure 7.28 Discrete-time filter for differentiator

Corresponding \(H_d(e^{j\Omega})\)

One practical application of this framework is designing a digital differentiator. If we want our overall system to compute the derivative of an analog signal, we know its ideal continuous-time frequency response is jω. By mapping this to the discrete-time domain using the frequency scaling, we get the desired frequency response for our digital filter. This Hd(e^jΩ) can then be approximated using digital filter design techniques, giving us a digital system that performs differentiation on analog signals.

Example: Impulse Response of Digital Differentiator

• Consider input \(x_c(t) = \frac{\sin(\pi t/T)}{\pi t}\) (a sinc function, band-limited).
• Sampled input \(x_d[n] = x_c(nT) = \frac{1}{T}\delta[n]\).
• The continuous-time derivative is \(y_c(t) = \frac{d}{dt} x_c(t) = \frac{\cos(\pi t/T)}{T t} - \frac{\sin(\pi t/T)}{\pi t^2}\).
• The discrete-time output \(y_d[n] = y_c(nT)\) is: \[ y_d[n] = \begin{cases} \frac{(-1)^n}{nT^2}, & n \neq 0 \\ 0, & n=0 \end{cases} \]
• Since \(x_d[n]\) is a scaled impulse, \(y_d[n]\) is the impulse response of the discrete-time filter: \[ h_d[n] = \begin{cases} \frac{(-1)^n}{nT}, & n \neq 0 \\ 0, & n=0 \end{cases} \]

Tip

This example demonstrates how to find the impulse response of a discrete-time filter that implements a continuous-time operation by analyzing its response to a specific band-limited input.

This example shows a concrete way to derive the impulse response of our digital differentiator. By feeding a specific band-limited input – a sinc function, whose samples are a simple impulse – we can observe the output. The output samples, yd[n], directly represent the impulse response hd[n] of the discrete-time filter. This hd[n] is an infinite impulse response (IIR) filter, indicating that an ideal differentiator cannot be perfectly realized with a finite number of taps, a common theme in digital filter design.

Application: Half-Sample Delay

• Goal: Implement a continuous-time delay: \(y_c(t) = x_c(t - \Delta)\).
• Continuous-Time Delay: \(H_c(j\omega) = e^{-j\omega\Delta}\).
• Band-Limited Delay: \[ H_c(j\omega) = \begin{cases} e^{-j\omega\Delta}, & |\omega| < \omega_c \\ 0, & \text{otherwise} \end{cases} \]
• Corresponding Discrete-Time Filter: \[ H_d(e^{j\Omega}) = e^{-j\Omega\Delta/T}, \quad |\Omega| < \pi \]
• If \(\Delta/T\) is an integer, \(y_d[n] = x_d[n - \Delta/T]\) (simple shift).
• If \(\Delta/T\) is not an integer (e.g., \(\Delta/T = 1/2\) for a half-sample delay), \(y_d[n]\) requires band-limited interpolation to effectively sample the shifted continuous signal.

Application: Half-Sample Delay

Figure 7.29(a) CT delay magnitude and phase

Ideal \(H_c(j\omega)\)

Figure 7.29(b) DT delay magnitude and phase

Corresponding \(H_d(e^{j\Omega})\)

Another powerful application is implementing time delays. While integer delays are straightforward in discrete time, fractional delays, like a half-sample delay, are more complex. The theory allows us to define the discrete-time filter’s frequency response Hd(e^jΩ) to achieve any arbitrary delay Δ. When Δ is not an integer multiple of T, the discrete-time output yd[n] effectively represents samples taken at fractional time points, which necessitates interpolation. This is incredibly useful in applications like audio effects, precise synchronization, or beamforming.

Interactive Demo: Half-Sample Delay

Visualize a continuous signal, its samples, and the effect of a fractional delay.

viewof delay_fraction = Inputs.range([0, 1], {value: 0.5, step: 0.05, label: "Delay (fraction of sample period)"});
viewof signal_type = Inputs.select(["Sine Wave", "Square Wave"], {value: "Sine Wave", label: "Signal Type"});

In this demo, the blue line is your original continuous signal, and the red dots are its discrete samples. The green dashed line shows what the continuous signal would look like if it were delayed by the fraction you select. The purple ‘x’ marks are the crucial part: they represent the discrete samples of that delayed continuous signal. Notice how these purple ‘x’ marks don’t necessarily align with the original red sample points. This highlights that to achieve a fractional delay in the discrete-time domain, the system needs to effectively “interpolate” the values that would exist between the original sample points.

Sampling of Discrete-Time Signals

• Analogy: Similar principles apply to sampling discrete-time signals.
• Process: Impulse-train sampling of a discrete-time signal \(x[n]\).
  • Produces \(x_p[n]\) where \(x_p[n] = x[n]\) at \(n=kN\) and \(0\) otherwise.
  • \(N\) is the sampling period (number of samples skipped).
• Frequency Domain: \[ X_p(e^{j\omega}) = \frac{1}{N} \sum_{k=0}^{N-1} X(e^{j(\omega - k\omega_s)}) \]
  • Here, \(\omega_s = 2\pi/N\) is the discrete-time sampling frequency.
• Aliasing: Occurs if \(\omega_s < 2\omega_M\), where \(\omega_M\) is the highest frequency in \(X(e^{j\omega})\) within \([0, \pi]\).
• Reconstruction: Lowpass filter with gain \(N\) and cutoff \(\omega_s/2\).

The concept of sampling isn’t limited to continuous-time signals. We can also sample discrete-time signals. The mechanics are very similar: we create a new sequence xp[n] by keeping x[n] at integer multiples of a sampling period N and setting all other values to zero. In the frequency domain, this again results in periodic replicas of the original discrete-time spectrum. Just like with continuous-time signals, if N is too small (meaning ωs is too low), aliasing will occur, and the original spectrum cannot be perfectly recovered.

Discrete-Time Decimation (Downsampling)

• Problem with \(x_p[n]\): It contains many zeros, which are inefficient for storage or transmission.
• Decimation (Downsampling): Creates a new sequence \(x_b[n]\) by extracting only the non-zero samples of \(x_p[n]\). \[ x_b[n] = x_p[nN] = x[nN] \]
  • Effectively reduces the sampling rate by a factor of \(N\).
• Frequency Domain: The spectrum of the decimated signal \(X_b(e^{j\omega})\) is a frequency-scaled version of \(X_p(e^{j\omega})\): \[ X_b(e^{j\omega}) = X_p(e^{j\omega/N}) \]
  • Decimation “stretches” the spectrum of the original sequence over a larger portion of the frequency band \([-\pi, \pi]\).
• Use Case: If the original continuous-time signal was oversampled, or if filtering reduced its bandwidth, decimation can reduce the data rate without aliasing.

Discrete-Time Decimation (Downsampling)

Figure 7.34 Relationship between xp[n] and xb[n]

Sampling vs. Decimation

Figure 7.35 Frequency-domain illustration of sampling and decimation

Effect of decimation in frequency domain

While impulse-train sampling gives us xp[n], it’s not efficient due to all the zeros. Decimation, or downsampling, addresses this by simply discarding those zeros, effectively compressing the sequence. In the frequency domain, this has the effect of stretching out the original signal’s spectrum to fill a larger portion of the discrete-time frequency range. This is useful when you have an oversampled signal or after you’ve filtered a signal to reduce its bandwidth, allowing you to reduce the data rate without losing information.

Discrete-Time Interpolation (Upsampling)

• Reverse of Decimation: Increases the effective sampling rate of a discrete-time sequence.
• Process to Upsample by a factor of \(L\):
  1. Zero-Insertion: Insert \(L-1\) zeros between each sample of the input sequence \(x[n]\) to create a zero-padded sequence \(x_p[n]\).
  2. Lowpass Filtering: Pass \(x_p[n]\) through a lowpass filter to smooth out the inserted zeros and reconstruct the interpolated sequence \(y[n]\).

Caution

The lowpass filter is crucial to remove the spectral images introduced by zero-insertion and prevent unwanted high-frequency components in the upsampled signal.

Upsampling is the inverse operation of decimation. It’s used when you need to increase the sampling rate of a discrete-time signal. This is done by inserting zeros between existing samples, which in the frequency domain creates copies of the original spectrum. To get a smooth, correctly interpolated signal, you then pass this zero-padded sequence through a lowpass filter. This filter removes the unwanted spectral images, leaving only the desired baseband component, effectively interpolating the signal to a higher sampling rate. This is common in digital audio for sample rate conversion, or in image processing for scaling.

Example: Combined Decimation and Interpolation

• Sometimes, a single decimation or interpolation factor isn’t enough, or a non-integer factor is desired.
• Scenario (Example 7.5): A sequence \(x[n]\) with spectrum \(X(e^{j\omega})\) that is zero for \(2\pi/9 \le |\omega| \le \pi\).
  • Maximum decimation without aliasing is \(N=4\).
  • However, the spectrum after decimation by 4 still has unused bandwidth.
• Achieving non-integer rate change (e.g., \(9/2\)):
  1. Upsample by a factor of \(L=2\).
  2. Downsample by a factor of \(M=9\).
  • Overall rate change: \(L/M = 2/9\).
  • This allows the spectrum to be stretched to fill the entire frequency band \([-\pi, \pi]\) without aliasing.

Example: Combined Decimation and Interpolation

Figure 7.38(a) Spectrum of x[n]

Original \(X(e^{j\omega})\)

Figure 7.38(d) Spectrum after upsampling by 2 and downsampling by 9

Final spectrum after rate change by 2/9

Note

This technique is vital in multi-rate signal processing, allowing efficient conversion between different sampling rates.

Sometimes, we need to change the sampling rate by a non-integer factor, or we need to optimize the bandwidth usage more precisely. This is where combining upsampling and downsampling becomes powerful. By first upsampling by an integer factor L and then downsampling by an integer factor M, we can achieve an arbitrary rational rate change L/M. This is crucial in applications like converting audio between different standard sample rates (e.g., 44.1 kHz to 48 kHz), or in communication systems for matching data rates between different components.

Summary

• Sampling Theorem: Foundation for converting continuous to discrete signals, requiring \(\omega_s > 2\omega_M\) for perfect reconstruction.
• Aliasing: Distortion caused by undersampling (\(\omega_s < 2\omega_M\)), where higher frequencies are “folded” into lower frequencies, leading to incorrect reconstruction.
  • Can be exploited in applications like the stroboscope effect.
• Discrete-Time Processing of Continuous-Time Signals: A powerful paradigm involving C/D conversion, discrete-time filtering, and D/C conversion.
  • Allows implementation of continuous-time LTI systems digitally.
  • Applications include digital differentiators and fractional delays.
• Discrete-Time Sampling:
  • Decimation (Downsampling): Reduces data rate by extracting every \(N\)th sample, efficient for oversampled or band-limited signals.
  • Interpolation (Upsampling): Increases data rate by inserting zeros and lowpass filtering, essential for sample rate conversion.

Important

Understanding these concepts is fundamental for designing and analyzing virtually all modern digital signal processing systems in ECE.

To summarize, we’ve covered the critical aspects of sampling today. The sampling theorem dictates the rules for perfect reconstruction. When these rules are broken, aliasing occurs, leading to distortion or, in some cases, interesting applications like the stroboscope. We then saw how these sampling principles enable us to process continuous-time signals using the flexibility of discrete-time systems, leading to applications like digital differentiators and fractional delays. Finally, we extended sampling to discrete-time signals with decimation and interpolation, which are crucial for efficient data handling and sample rate conversion. These concepts are not just theoretical; they are the bedrock of digital audio, video, communications, control systems, and countless other ECE applications.