GATE EE

Sampling Theory for Signals & Systems: GATE Quick Notes

Lecture Notes

SEC 01

Sampling Fundamentals

SEC 02

Introduction to Sampling

1Introduction to Sampling
1What is Sampling?

Converting a continuous-time signal \(x(t)\) into a discrete-time sequence \(x[n]\) by taking samples at regular intervals.

1Mathematical Representation
\[\begin{aligned} x[n] &= x(nT_s) \quad \text{for } n = 0, \pm 1, \pm 2, \ldots \\ T_s &= \text{Sampling period} \\ f_s &= \frac{1}{T_s} = \text{Sampling frequency} \\ \omega_s &= 2\pi f_s = \text{Sampling frequency (rad/s)} \end{aligned}\]
1Sampling Process
\[\begin{aligned} x_s(t) = x(t) \cdot p(t) = x(t) \sum_{n=-\infty}^{\infty} \delta(t - nT_s) \end{aligned}\]
where \(p(t)\) is the sampling function (impulse train).
SEC 03

Impulse Train Sampling

1Impulse Train Sampling
1Sampling Function
\[\begin{aligned} p(t) &= \sum_{n=-\infty}^{\infty} \delta(t - nT_s) \\ P(\omega) &= \frac{2\pi}{T_s} \sum_{k=-\infty}^{\infty} \delta(\omega - k\omega_s) \end{aligned}\]
1Sampled Signal in Frequency Domain
\[\begin{aligned} X_s(\omega) &= \frac{1}{2\pi}[X(\omega) * P(\omega)] \\ &= \frac{1}{T_s} \sum_{k=-\infty}^{\infty} X(\omega - k\omega_s) \end{aligned}\]
1Key Insight

The spectrum of the sampled signal consists of the original spectrum repeated at intervals of \(\omega_s\).

SEC 04

Nyquist Sampling Theorem

SEC 05

Nyquist Sampling Theorem

1Nyquist Sampling Theorem
1Statement

A band-limited signal \(x(t)\) with maximum frequency \(f_m\) can be perfectly reconstructed from its samples if the sampling frequency satisfies:

\[\begin{aligned} f_s \geq 2f_m \end{aligned}\]
1Key Definitions

  • Nyquist Rate: \(f_N = 2f_m\) (minimum sampling rate)

  • Nyquist Frequency: \(f_N/2 = f_m\) (maximum signal frequency)

  • Nyquist Interval: \(T_N = 1/f_N = 1/(2f_m)\) (maximum sampling period)

1Conditions for Perfect Reconstruction

  1. Signal must be band-limited: \(X(\omega) = 0\) for \(|\omega| > \omega_m\)

  2. Sampling rate: \(\omega_s > 2\omega_m\)

  3. Ideal reconstruction filter required

SEC 06

Nyquist Theorem - Frequency Domain View

1Nyquist Theorem - Frequency Domain View
1Case 1: \(\omega_s > 2\omega_m\) (No Aliasing)

  • Spectral replicas do not overlap

  • Perfect reconstruction possible

  • \(X_s(\omega) = \frac{1}{T_s}X(\omega)\) for \(|\omega| < \omega_s/2\)

1Case 2: \(\omega_s < 2\omega_m\) (Aliasing)

  • Spectral replicas overlap

  • Information loss occurs

  • Perfect reconstruction impossible

1Case 3: \(\omega_s = 2\omega_m\) (Critical Sampling)

  • Spectral replicas just touch

  • Theoretically perfect reconstruction possible

  • Practically challenging due to ideal filter requirements

SEC 07

Aliasing

SEC 08

Aliasing Phenomenon

1Aliasing Phenomenon
1Definition

Aliasing occurs when the sampling rate is insufficient (\(f_s < 2f_m\)), causing high-frequency components to appear as lower frequencies in the sampled signal.

1Mathematical Description

For a sinusoid \(x(t) = A\cos(2\pi f_0 t)\) sampled at rate \(f_s\):

\[\begin{aligned} x[n] &= A\cos(2\pi f_0 nT_s) = A\cos\left(\frac{2\pi f_0 n}{f_s}\right) \end{aligned}\]

If \(f_0 > f_s/2\), the sampled signal appears as:

\[\begin{aligned} x[n] = A\cos\left(\frac{2\pi f_a n}{f_s}\right) \end{aligned}\]
where \(f_a\) is the alias frequency.
SEC 09

Alias Frequency Calculation

1Alias Frequency Calculation
1Alias Frequency Formula

For a frequency \(f_0\) sampled at rate \(f_s\):

\[\begin{aligned} f_a = |f_0 - kf_s| \end{aligned}\]
where \(k\) is chosen such that \(0 \leq f_a \leq f_s/2\).
1Alternative Formula
\[\begin{aligned} f_a = \begin{cases} f_0 - kf_s & \text{if } kf_s < f_0 < (k+0.5)f_s \\ (k+1)f_s - f_0 & \text{if } (k+0.5)f_s < f_0 < (k+1)f_s \end{cases} \end{aligned}\]
for integer \(k \geq 0\).
1Example

If \(f_0 = 1.2\) kHz and \(f_s = 1\) kHz: \(f_a = |1.2 - 1| = 0.2\) kHz

SEC 10

Types of Aliasing

1Types of Aliasing
1Temporal Aliasing

  • High-frequency signals appear as low-frequency

  • Common in digital signal processing

  • Example: Wagon wheel effect in movies

1Spectral Aliasing

  • Frequency domain overlapping

  • Loss of original frequency information

  • Irreversible distortion

1Practical Examples

  • Audio sampling: Need \(f_s > 40\) kHz for 20 kHz audio

  • Video sampling: Frame rate vs. motion frequency

  • Radar systems: Doppler frequency aliasing

SEC 11

Anti-Aliasing

SEC 12

Anti-Aliasing Techniques

1Anti-Aliasing Techniques
1Pre-filtering (Anti-Aliasing Filter)

  • Low-pass filter before sampling

  • Cutoff frequency: \(f_c \leq f_s/2\)

  • Removes frequency components above Nyquist frequency

1Anti-Aliasing Filter Requirements
\[\begin{aligned} H(f) = \begin{cases} 1 & \text{for } |f| \leq f_m \\ 0 & \text{for } |f| \geq f_s/2 \end{cases} \end{aligned}\]
1Practical Considerations

  • Ideal brick-wall filter not realizable

  • Transition band: \(f_m < f < f_s/2\)

  • Guard band: \(f_g = f_s/2 - f_m\)

  • Higher sampling rate reduces filter complexity

SEC 13

Oversampling

1Oversampling
1Definition

Sampling at a rate much higher than the Nyquist rate:

\[\begin{aligned} f_s >> 2f_m \end{aligned}\]
1Advantages

  • Relaxed anti-aliasing filter requirements

  • Better SNR performance

  • Easier analog filter design

  • Noise shaping possibilities

1Oversampling Ratio
\[\begin{aligned} \text{OSR} = \frac{f_s}{2f_m} \end{aligned}\]
1Applications

  • Delta-sigma modulators

  • High-resolution audio systems

  • Medical imaging systems

SEC 14

Signal Reconstruction

SEC 15

Signal Reconstruction

1Signal Reconstruction
1Reconstruction Process

Converting discrete-time samples back to continuous-time signal:

\[\begin{aligned} x_r(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot h_r(t - nT_s) \end{aligned}\]
where \(h_r(t)\) is the reconstruction filter impulse response.
1Ideal Reconstruction

For perfect reconstruction (when Nyquist criterion is satisfied):

\[\begin{aligned} H_r(\omega) = \begin{cases} T_s & \text{for } |\omega| \leq \omega_s/2 \\ 0 & \text{for } |\omega| > \omega_s/2 \end{cases} \end{aligned}\]
1Ideal Reconstruction Filter
\[\begin{aligned} h_r(t) = \frac{\sin(\omega_s t/2)}{\omega_s t/2} = \frac{1}{2} \text{sinc}\left(\frac{t}{2T_s}\right) \end{aligned}\]
SEC 16

Whittaker-Shannon Interpolation Formula

1Whittaker-Shannon Interpolation Formula
1Perfect Reconstruction Formula

When sampling theorem conditions are met:

\[\begin{aligned} x(t) = \sum_{n=-\infty}^{\infty} x(nT_s) \cdot \frac{\sin[\pi(t-nT_s)/T_s]}{\pi(t-nT_s)/T_s} \end{aligned}\]
1Simplified Form
\[\begin{aligned} x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot \text{sinc}\left(\frac{t-nT_s}{T_s}\right) \end{aligned}\]
1Key Properties

  • Each sample contributes a sinc function

  • Sinc functions are orthogonal at sampling instants

  • Perfect reconstruction requires infinite sum

  • Practical systems use finite approximations

SEC 17

Practical Reconstruction Methods

1Practical Reconstruction Methods
1Zero-Order Hold (ZOH)
\[\begin{aligned} x_r(t) = x[n] \quad \text{for } nT_s \leq t < (n+1)T_s \end{aligned}\]

  • Simplest reconstruction

  • Creates staircase waveform

  • Significant high-frequency distortion

1First-Order Hold (Linear Interpolation)
\[\begin{aligned} x_r(t) = x[n] + \frac{t-nT_s}{T_s}(x[n+1] - x[n]) \end{aligned}\]

  • Linear interpolation between samples

  • Better than ZOH but still imperfect

  • Reduced high-frequency content

1Higher-Order Interpolation

  • Cubic spline interpolation

  • Windowed sinc interpolation

  • Better approximation to ideal reconstruction

SEC 18

Practical Sampling Systems

SEC 19

Sample-and-Hold (S&H) Circuits

1Sample-and-Hold (S&H) Circuits
1Purpose

  • Hold sample value constant during A/D conversion

  • Prevent aperture error

  • Provide time for conversion process

1S&H Transfer Function
\[\begin{aligned} H_{SH}(\omega) = \frac{\sin(\omega T_h/2)}{\omega T_h/2} e^{-j\omega T_h/2} \end{aligned}\]
where \(T_h\) is the hold time.
1Aperture Effects

  • Aperture Time: Time to acquire sample

  • Aperture Jitter: Variation in sampling instant

  • Aperture Error: Signal change during acquisition

SEC 20

Quantization in Sampling

1Quantization in Sampling
1Quantization Process

Converting continuous amplitude to discrete levels:

\[\begin{aligned} x_q[n] = Q(x[n]) \end{aligned}\]
1Uniform Quantization
\[\begin{aligned} \Delta &= \frac{x_{\max} - x_{\min}}{2^b} \quad \text{(Quantization step)} \\ b &= \text{Number of bits} \end{aligned}\]
1Quantization Error
\[\begin{aligned} e_q[n] &= x_q[n] - x[n] \\ -\frac{\Delta}{2} &\leq e_q[n] \leq \frac{\Delta}{2} \end{aligned}\]
1Signal-to-Quantization-Noise Ratio
\[\begin{aligned} \text{SQNR} = 6.02b + 1.76 \text{ dB} \end{aligned}\]
for uniform quantization with \(b\) bits.
SEC 21

Multirate Signal Processing

SEC 22

Decimation (Downsampling)

1Decimation (Downsampling)
1Definition

Reducing the sampling rate by factor \(M\):

\[\begin{aligned} y[n] = x[Mn] \end{aligned}\]
1Frequency Domain Effect
\[\begin{aligned} Y(\omega) = \frac{1}{M} \sum_{k=0}^{M-1} X\left(\frac{\omega - 2\pi k}{M}\right) \end{aligned}\]
1Anti-Aliasing for Decimation

  • Low-pass filter before decimation

  • Cutoff frequency: \(\omega_c = \pi/M\)

  • Prevents aliasing in downsampled signal

1Decimation Process
\[\begin{aligned} x[n] \xrightarrow{LPF} w[n] \xrightarrow{\downarrow M} y[n] \end{aligned}\]
SEC 23

Interpolation (Upsampling)

1Interpolation (Upsampling)
1Definition

Increasing the sampling rate by factor \(L\):

\[\begin{aligned} w[n] = \begin{cases} x[n/L] & \text{if } n = 0, \pm L, \pm 2L, \ldots \\ 0 & \text{otherwise} \end{cases} \end{aligned}\]
1Frequency Domain Effect
\[\begin{aligned} W(\omega) = X(L\omega) \end{aligned}\]
1Anti-Imaging Filter

  • Low-pass filter after upsampling

  • Cutoff frequency: \(\omega_c = \pi/L\)

  • Removes spectral images

1Interpolation Process
\[\begin{aligned} x[n] \xrightarrow{\uparrow L} w[n] \xrightarrow{LPF} y[n] \end{aligned}\]
SEC 24

Important Formulas Summary

SEC 25

Key Formulas - Sampling Theory

1Key Formulas - Sampling Theory
1Fundamental Relations
\[\begin{aligned} \text{Nyquist Rate: } & f_s \geq 2f_m \\ \text{Sampling: } & x_s(t) = x(t) \sum_{n=-\infty}^{\infty} \delta(t - nT_s) \\ \text{Spectrum: } & X_s(\omega) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} X(\omega - k\omega_s) \\ \text{Reconstruction: } & x(t) = \sum_{n=-\infty}^{\infty} x[n] \text{sinc}\left(\frac{t-nT_s}{T_s}\right) \\ \text{Alias Frequency: } & f_a = |f_0 - kf_s| \\ \text{SQNR: } & \text{SQNR} = 6.02b + 1.76 \text{ dB} \end{aligned}\]
SEC 26

Common GATE Problems

1Common GATE Problems
1Typical Question Types

  1. Finding minimum sampling rate for given signal

  2. Calculating alias frequencies

  3. Determining reconstruction filter specifications

  4. Multirate system design

  5. Quantization noise calculations

1Problem-Solving Tips

  • Always identify maximum frequency component

  • Check for aliasing conditions

  • Consider practical filter limitations

  • Remember guard bands in real systems

  • Use frequency domain analysis for complex signals