Bandpass Sampling Theorem: Derivation and Application

Hello! Welcome back.

In our previous lessons, we established the foundations of sampling theory, culminating in the Nyquist-Shannon theorem: to perfectly reconstruct a low-pass signal, we must sample at a rate greater than twice its highest frequency component ( $f_{s} > 2 f_{ma x}$ ). We also derived the critical scaling relationship, $ω = Ω T_{s}$ , that connects the continuous and discrete frequency domains.

This works perfectly for signals whose content starts at or near DC. But consider a typical radio signal: it might have a bandwidth of only 200 kHz, but be centered around a carrier frequency of 100 MHz. The standard Nyquist criterion would demand a sampling rate over 200 MHz, which is computationally expensive and often impractical. Today, we address this challenge head-on. Our goal is to derive and apply the bandpass sampling theorem, a more general and efficient approach for sampling signals that occupy a specific frequency band away from DC.

1. The Core Idea: "Intelligent Aliasing"

The standard sampling theorem is designed to avoid aliasing completely. Bandpass sampling, also known as undersampling or sub-sampling, takes a different view. It recognizes that the spectral replicas created by sampling are not inherently a problem, as long as one of them can be captured cleanly, without overlapping with any others.

The goal is to choose a sampling frequency $f_{s}$ low enough to be efficient, but in such a clever way that a complete, uncorrupted spectral copy of our bandpass signal lands perfectly in the baseband region, $[0, f_{s} /2]$ . This process is sometimes called "intelligent aliasing by design."

To visualize this, first watch a short conceptual introduction.

A/D and D/A Sampling Theory

The video 'A/D and D/A Sampling Theory' provides a great introduction to this concept, which it calls 'undersampling.' It introduces the idea of Nyquist Zones, which are frequency bands of width f_s/2.

Please watch from 16:05 to 19:21 and the summary from 20:26 to 21:02. Focus on these key ideas: The core rule: all signal energy must fall within a single Nyquist Zone. The benefit: it allows a lower f_s for high-frequency signals, by leveraging the fact that the sampling rate must be twice the signal's bandwidth, not necessarily its highest frequency. The framing of this technique as 'intelligent aliasing'.

This idea that different high-frequency bands can all "fold" down into the same baseband upon sampling is the heart of the matter. The key is to ensure that our desired band folds down without another unwanted band folding on top of it.

Multiple bands aliasing to baseband

This figure from the provided paper, "The Sampling Theorem and the Bandpass Theorem", illustrates the core concept. It shows the spectra of four different continuous-time signals, each in a different frequency band. When sampled at a rate of 2\pi (in this normalized example), all four produce the exact same sampled sequence because their spectra alias, or 'fold', into the same baseband interval [-\pi, \pi]. Our task is to choose a sampling rate such that our signal of interest does this cleanly.

2. Deriving the Bandpass Sampling Theorem

Now, let's derive the mathematical conditions that guarantee an alias-free result. We want to find the allowed ranges for the sampling frequency, $f_{s}$ , for a bandpass signal with a spectrum non-zero only in the interval $[f_{L}, f_{U}]$ (and its negative counterpart). The signal bandwidth is $B = f_{U} - f_{L}$ .

A highly intuitive way to derive these conditions is to visualize the spectral replicas and determine the constraints needed to prevent them from overlapping.

Bandpass Sampling | Sampling Theorem for Bandpass Sampling | Tutorial

The video 'Bandpass Sampling | Tutorial' provides a clear, step-by-step graphical derivation of the conditions for alias-free bandpass sampling. This visual approach is excellent for building a deep understanding.

Please watch the segment from 06:05 to 13:20. Don't worry about memorizing every step, but focus on the logic. The presenter considers two adjacent spectral replicas and derives two inequalities that must be met to avoid overlap. This process ultimately leads to the final conditions on the sampling frequency f_s.

The geometric derivation in the video, along with more formal proofs, leads to the central result of the bandpass sampling theorem. This result is stated concisely in the paper by D.S.G. Pollock.

The Sampling Theorem and the Bandpass Theorem

Now, let's look at the formal mathematical statement of the theorem from the paper 'The Sampling Theorem and the Bandpass Theorem'.

Please read the final part of the paper, starting from the section 'The Bandpass Sampling Theory' to the end. The key takeaway is the inequality (28), which defines the allowed sampling rates.

The fundamental condition derived in these resources is that for some integer $n \geq 1$ , the sampling frequency $f_{s}$ must satisfy:

\frac{2 f _{U}}{n} \leq f_{s} \leq \frac{2 f _{L}}{n - 1}

This inequality defines a set of disjoint frequency ranges, or "windows," where you can set your sampling frequency to successfully capture the bandpass signal.

What does the integer $n$ represent?

It corresponds to the Nyquist zone in which the signal's band $[f_{L}, f_{U}]$ resides.
For this inequality to define a valid, non-empty range (i.e., for the lower bound not to exceed the upper bound), $n$ cannot be arbitrarily large. A little algebra shows that we must have $n \leq \frac{f _{U}}{B}$ .
The lowest possible sampling rate is always at least $2 B$ .

So, the procedure to find all possible alias-free sampling rates is:

Identify the signal's lower and upper frequencies, $f_{L}$ and $f_{U}$ , and calculate its bandwidth $B = f_{U} - f_{L}$ .
Determine the maximum possible value for the integer $n$ using $n_{ma x} = ⌊ \frac{f _{U}}{B} ⌋$ .
For each integer $n$ from $1$ to $n_{ma x}$ , calculate the corresponding allowed range for $f_{s}$ using the formula.

3. Applying the Theorem: An Example

Let's solidify this with an example. Suppose we have a signal with a bandwidth of 2 MHz, centered at 21 MHz.

Upper Frequency: $f_{U} = 22$ MHz
Lower Frequency: $f_{L} = 20$ MHz
Bandwidth: $B = f_{U} - f_{L} = 2$ MHz

The standard Nyquist theorem would require $f_{s} > 2 f_{U} = 44$ MHz. Let's see what the bandpass theorem allows.

Find $n_{ma x}$ :
$n_{ma x} = ⌊ \frac{f _{U}}{B} ⌋ = ⌊ \frac{22}{2} ⌋ = 11$
So we can test integers $n$ from 1 to 11.
Calculate the allowed ranges for $f_{s}$ :
- n = 1: $\frac{2 ( 22 )}{1} \leq f_{s} \leq \frac{2 ( 20 )}{0} \to f_{s} \geq 44$ MHz. This is the standard Nyquist case.
- n = 2: $\frac{2 ( 22 )}{2} \leq f_{s} \leq \frac{2 ( 20 )}{1} \to 22 MHz \leq f_{s} \leq 40$ MHz.
- ...
- n = 10: $\frac{2 ( 22 )}{10} \leq f_{s} \leq \frac{2 ( 20 )}{9} \to 4.4 MHz \leq f_{s} \leq 4.444$ MHz. A very narrow but valid range!
- n = 11: $\frac{2 ( 22 )}{11} \leq f_{s} \leq \frac{2 ( 20 )}{10} \to 4.0 MHz \leq f_{s} \leq 4.0$ MHz. This means only $f_{s} = 4.0$ MHz is allowed for this $n$ .

This is a remarkable result! We can perfectly sample a 22 MHz signal using a sampling rate as low as 4.4 MHz, a factor of 10 reduction from the naive approach. This is why bandpass sampling is so crucial in fields like radio communications and software-defined radio.

The allowed and forbidden regions for $f_{s}$ are often visualized in a plot, like the one shown in the resources you've reviewed.

4. Important Considerations

Spectral Inversion

There is one crucial detail when you recover the signal from its undersampled version. The aliased copy in the baseband might be spectrally inverted (flipped).

If the integer $n$ (the Nyquist zone index) is odd, the spectrum in the baseband has the same orientation as the original.
If $n$ is even, the spectrum is inverted.

This is a predictable effect that can be easily corrected in software if needed, for instance by multiplying the time-domain signal by $(- 1)^{k}$ , but it's essential to be aware of.

Sampling bandpass signals

The article 'Sampling bandpass signals' by Josef Hoffmann provides a clear explanation of spectral inversion, along with some practical formulas and context.

Please read section 1.2 to see the discussion of how even and odd values of the zone index (called NZ in the article) lead to inversion. Also, I encourage you to read the short comment exchange with Richard Lyons at the end (section 1.6). It's a nice historical note from a prominent figure in the DSP community that captures the excitement of discovering this 'always true' sampling rule.

The "always true" rule mentioned by Richard Lyons is that to preserve all information, the sampling rate must be at least twice the bandwidth of the analog signal ( $f_{s} \geq 2 B$ ). The standard Nyquist theorem for lowpass signals is just a special case of this more general principle.

Test your understanding!

Consider a bandpass signal with a spectrum between 15 kHz and 25 kHz.

$f_{L} = 15$ kHz
$f_{U} = 25$ kHz
$B = 10$ kHz

Using the bandpass sampling theorem, determine which of the following sampling frequencies would allow for the signal to be recovered without aliasing:
a) $f_{s} = 22$ kHz
b) $f_{s} = 28$ kHz
c) $f_{s} = 55$ kHz

For any valid sampling rate, will the resulting baseband spectrum be inverted?

Show answer

First, let's find the maximum value for $n$ :
$n_{ma x} = ⌊ f_{U} / B ⌋ = ⌊ 25/10 ⌋ = 2$ . So we only need to check the cases for $n = 1$ and $n = 2$ .

Case n = 1 (Odd Zone):
$\frac{2 ( 25 )}{1} \leq f_{s} ⟹ f_{s} \geq 50 kHz$
This is the standard low-pass sampling condition.
- c) $f_{s} = 55$ kHz is a valid sampling frequency.
- Since $n = 1$ is odd, the spectrum will not be inverted.
Case n = 2 (Even Zone):
$\frac{2 ( 25 )}{2} \leq f_{s} \leq \frac{2 ( 15 )}{2 - 1} ⟹ 25 kHz \leq f_{s} \leq 30 kHz$
- b) $f_{s} = 28$ kHz is a valid sampling frequency.
- Since $n = 2$ is even, the spectrum will be inverted.
a) $f_{s} = 22$ kHz does not fall into any of the allowed ranges, so it is not a valid sampling frequency and would result in aliasing.

Conclusion

Today, we've unlocked a powerful and efficient technique that overcomes the apparent limitations of the basic sampling theorem. By embracing "intelligent aliasing," we can sample high-frequency, band-limited signals at rates dramatically lower than their highest frequency components would suggest.

Key Takeaways:

The bandpass sampling theorem allows for efficient sampling of bandpass signals by choosing a sampling rate $f_{s}$ that cleanly aliases a copy of the signal's spectrum into the baseband.
The fundamental requirement is that the sampling rate must be at least twice the signal's bandwidth ( $f_{s} \geq 2 B$ ).
The permissible sampling rates fall into distinct ranges, defined by the formula $\frac{2 f _{U}}{n} \leq f_{s} \leq \frac{2 f _{L}}{n - 1}$ for integers $n$ from 1 to $⌊ f_{U} / B ⌋$ .
Sampling from an even-numbered Nyquist zone ( $n$ is even) results in a spectrally inverted signal in the baseband, a predictable and correctable effect.

Preview of the Next Module:

We have now completed our exploration of the bridge between the analog and digital worlds. We understand how to sample, the perils of aliasing, the role of filters, and the relationship between continuous and discrete frequencies.

In the next module, we will begin our deep dive into the analysis of discrete-time signals and systems. We will start by tracing the history of the z-transform, a powerful generalization of the Discrete-Time Fourier Transform, which serves the same role for discrete-time systems that the Laplace transform serves for analog systems.

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