An FM radio signal occupies the band from 99.9 MHz to 100.1 MHz (200 kHz bandwidth). What is the minimum sampling rate required under the bandpass sampling theorem?
A200.2 MHz — slightly above twice the highest frequency in the signal
B400 kHz — twice the signal bandwidth
C100.1 MHz — equal to the highest frequency
DAny rate below 200.2 MHz works automatically, as long as it exceeds twice the bandwidth
The bandpass sampling theorem states that the minimum rate is 2B where B is the signal bandwidth, provided the rate is chosen carefully to avoid spectral overlap. With B = 200 kHz, the minimum is 400 kHz — a factor of 500× less than the naïve 200.2 MHz (2 × f_max). Option D is wrong because not any rate below 2f_max works: the rate must be selected so that spectral copies of the signal land in non-overlapping positions. The valid rates form discrete windows, and choosing carelessly within those windows can still cause aliasing.
Question 2 Multiple Choice
A bandpass signal is centered at 1 GHz with a bandwidth of 10 MHz. An engineer samples it at 25 MHz. The resulting digital samples contain a version of the signal at a much lower frequency. What has happened?
AThe signal was destroyed by aliasing because the rate is far below 2 GHz
BThe sampling process has downconverted the signal to a lower frequency by deliberately exploiting aliasing
CThe signal was destroyed because 25 MHz is not an integer fraction of 1 GHz
DThe signal is present at 1 GHz in the digital samples, but the ADC only stores it at 25 MHz
Bandpass sampling deliberately exploits the aliasing mechanism: the spectral copies produced by sampling fold the high-frequency signal down to a lower frequency slot. This is intentional downconversion — the sampled signal is a low-frequency replica of the original bandpass signal, ready for digital processing at a fraction of the original rate. Option A applies only if the copies overlap and destroy each other. If the sampling rate was chosen correctly (validating the spectral window conditions), the folded copy is a clean, recoverable version of the original signal.
Question 3 True / False
Bandpass sampling can achieve correct signal recovery even when sampling below the rate that would be required by the standard Nyquist theorem applied naïvely to the highest signal frequency.
TTrue
FFalse
Answer: True
This is exactly the point of bandpass sampling. The standard Nyquist condition (f_s ≥ 2f_max) is a sufficient condition for a lowpass signal starting at DC. For a bandpass signal with no energy near DC, a lower rate suffices because the spectral copies need only avoid overlapping each other — not avoid overlapping with DC content that doesn't exist. The true requirement is that no two spectral copies overlap, and for a narrow-bandwidth bandpass signal, this can be satisfied at rates as low as 2B, where B is the bandwidth.
Question 4 True / False
Any sampling rate that exceeds twice the signal bandwidth and is below twice the highest frequency will correctly recover a bandpass signal without aliasing.
TTrue
FFalse
Answer: False
This is the most common misconception. The valid sampling rates for a bandpass signal form specific windows — not a continuous range from 2B to 2f_max. Within a given window, careful selection is required to ensure that spectral copies land in non-overlapping positions. A rate that technically satisfies 2B < f_s < 2f_max but is chosen carelessly can still cause aliasing if the folded spectrum partially overlaps an adjacent copy. The condition requires calculating valid windows for each integer n and verifying that f_s places the folded spectrum cleanly within an unoccupied frequency slot.
Question 5 Short Answer
What does the true Nyquist sampling requirement actually state, and how does this differ from the common statement 'sample at twice the highest frequency'? Why does this distinction matter for bandpass signals?
Think about your answer, then reveal below.
Model answer: The true requirement is that the sampling rate must be high enough that the spectral copies created by sampling do not overlap each other. When you sample at rate f_s, the spectrum is repeated at every multiple of f_s. Overlap-free copies allow perfect reconstruction with a filter. For a lowpass signal from DC to B, avoiding overlap requires f_s ≥ 2B — which equals 2f_max since f_max = B. For a bandpass signal from f_low to f_high with bandwidth B = f_high - f_low, the copies only need to avoid each other (not avoid DC), so a rate of 2B can suffice — far below 2f_high. The distinction matters because it enables sampling a 1 GHz signal at 20 MHz rather than 2 GHz, reducing hardware cost and processing burden by 100×.
The 'twice the highest frequency' rule is a correct consequence of the true rule for lowpass signals, where f_max = bandwidth. Teaching it as the fundamental rule causes confusion when students encounter bandpass signals, leading to the misconception that you always need to sample at 2f_max. Understanding the true requirement — spectral copy non-overlap — immediately reveals when and how lower rates are valid.