Sampling and bandpass selection in AM systems An arbitrary signal m(t) has zero average value and is strictly band-limited to 3.2 kHz. It is uniformly sampled at a rate of 8 k samples/s (fs = 8 kHz). The sampled signal is then passed through an ideal band-pass filter centered at 32 kHz with a bandwidth of 6.4 kHz (i.e., it passes 28.8 to 35.2 kHz). What is the nature of the output from this band-pass filter?

Introduction / Context:
Sampling by an impulse train replicates a baseband spectrum around integer multiples of the sampling frequency. Selecting one of these replicated bands with a band-pass filter is a classic way to implement modulation by spectral translation. This question tests recognition of how an ideally sampled, band-limited signal appears after a narrow band-pass centered on a sampling harmonic.

Given Data / Assumptions:

• Baseband is m(t), zero mean, band-limited to 3.2 kHz.
• Sampling frequency fs = 8 kHz.
• Ideal band-pass filter centered at 32 kHz with bandwidth 6.4 kHz (passes 32 ± 3.2 kHz).
• Ideal impulse-train sampling (no aperture effects).

Concept / Approach:
Impulse sampling multiplies m(t) by a Dirac comb in time, which in frequency creates repeated spectra M(f) centered at integer multiples kfs. Around f = 4fs = 32 kHz, the band 32 ± 3.2 kHz is a faithful copy of the baseband spectrum. Because the replica contains upper and lower sidebands around 32 kHz but no discrete carrier at exactly 32 kHz (mean of m(t) is zero), the selected output is a double-sideband suppressed-carrier (DSB-SC) signal.

Step-by-Step Solution:
Replicas appear at k*fs = …, 24, 32, 40 kHz, etc.The filter passes only the 32 ± 3.2 kHz band (28.8–35.2 kHz).This band contains symmetric upper and lower sidebands of m(t) shifted to 32 kHz.No discrete carrier exists at 32 kHz because m(t) has zero average (no impulse at DC to translate).Therefore the output is AM-DSB suppressed-carrier.

Verification / Alternative check:
If the message had a DC component, impulse sampling would create a spectral line at multiples of fs; filtering at 32 kHz could pass a tone at 32 kHz (a residual “carrier”). Here, zero mean removes that tone, confirming suppressed carrier.

Why Other Options Are Wrong:
AM-DSB with carrier: Requires a discrete line at the center frequency; not present for zero-mean m(t).

AM-SSB with carrier: Single sideband would need asymmetrical filtering to remove one sideband, which is not the case.

Exponentially decaying sines: That description matches damped sinusoids, unrelated to ideal filtering of sampled spectra.

FM constant envelope: No frequency modulation mechanism is involved.

Common Pitfalls:
Confusing DSB-SC created by spectral replication with conventional product modulation; both yield identical spectra but the sampling origin often misleads students into expecting a residual carrier.

Final Answer:
AM-DSB signal with suppressed carrier