Sampling Theorem

The Sampling Theorem is one of the most fundamental results in signal processing and communications. It provides the theoretical basis for converting continuous-time analog signals into discrete-time digital signals without any loss of information, provided a critical condition on the sampling rate is satisfied. This result directly underpins all modern digital audio, communication, and instrumentation systems.

Core Concept: The Sampling Theorem

The Nyquist-Shannon Sampling Theorem states that a band-limited continuous-time signal with maximum frequency component fm can be perfectly reconstructed from its samples if the sampling frequency fs satisfies fs >= 2fm. The minimum sampling rate of 2fm is called the Nyquist rate, and the corresponding maximum sampling interval T = 1/(2fm) is called the Nyquist interval.

The physical intuition is straightforward. A signal with maximum frequency fm completes fm cycles per second. To capture each cycle accurately, at least two samples per cycle are needed, one for the positive peak region and one for the negative peak region. Sampling below this rate causes cycles to be underrepresented, leading to ambiguity in reconstruction.

A signal is called band-limited if its Fourier transform X(f) is zero for all frequencies above fm. Practical signals are approximately band-limited after passing through an anti-aliasing low-pass filter before the analog-to-digital conversion stage.

Mathematical Expression

Sampling a continuous-time signal x(t) with a train of impulses at interval T produces the sampled signal xs(t) = sum of x(nT)*delta(t - nT). The Fourier transform of xs(t) is Xs(f) = (1/T) * sum of X(f - k/T) for integer k. This result shows that the spectrum of the sampled signal is a periodic repetition of the original spectrum X(f) with period fs = 1/T.

If fs >= 2fm, the repeated spectral copies do not overlap. In this case, passing xs(t) through an ideal low-pass filter with cutoff at fm and gain T perfectly recovers X(f), and therefore x(t). If fs < 2fm, the spectral copies overlap and the original signal cannot be recovered. This overlap is called aliasing and represents an irreversible loss of information.

Practical Understanding

In digital audio, the human hearing range extends to approximately 20 kHz. This is why the CD audio standard uses a sampling rate of 44.1 kHz, which comfortably satisfies the Nyquist criterion for fm = 20 kHz. Similarly, telephony uses 8 kHz sampling for voice signals bandlimited to 4 kHz.

Before sampling any real-world signal, an anti-aliasing filter is applied. This is an analog low-pass filter with cutoff at fm that removes all frequency components above fm. Without this filter, high-frequency noise or signal components above fm would alias into the baseband and corrupt the sampled data.

Given:
A continuous signal x(t) = 3cos(2*pi*1000*t) + 2cos(2*pi*3000*t)
Find the minimum sampling rate and sampling interval.

Why this formula applies:
The signal has two frequency components: f1 = 1000 Hz, f2 = 3000 Hz.
Maximum frequency fm = 3000 Hz.
Apply Nyquist criterion.

Formula:
fs_min = 2 * fm
T_max = 1 / fs_min

Substitution:
fs_min = 2 * 3000 = 6000 Hz
T_max = 1 / 6000

Calculation:
fs_min = 6 kHz
T_max = 0.1667 ms

Final Answer:
Minimum sampling rate = 6000 Hz (6 kHz)
Maximum sampling interval = 0.1667 milliseconds

Exam Tip: The Nyquist rate is 2fm, not fm. A common GATE trap is confusing Nyquist rate with Nyquist frequency. The Nyquist frequency is fm (half the sampling rate), while the Nyquist rate is the minimum fs = 2fm required for the signal.

Quick Revision

• Sampling theorem: fs >= 2fm for perfect reconstruction of a band-limited signal.
• Nyquist rate = 2fm (minimum sampling frequency).
• Nyquist interval = 1/(2fm) (maximum time between samples).
• Sampled spectrum = periodic repetition of X(f) with period fs.
• Anti-aliasing filter applied before sampling to limit signal bandwidth to fm.
• If fs < 2fm, aliasing occurs and original signal cannot be recovered.
• Exam trap: Nyquist frequency (fm) and Nyquist rate (2fm) are different quantities.