Sampling Theorem and Signal Processing

Digital Signal Processing (DSP) Lecture 2 Notes

• A signal is a function conveying information about a phenomenon's behavior or attributes.
• Signals can be continuous or discrete.
• Continuous-time signals are defined for every value of time (t).
  • Example: x(t) = sin(2πft), where t is a continuous variable.
• Discrete-time signals are defined only at discrete points in time.
  • Example: x[n] = sin(2πfnT), where n is an integer, and T is the sampling period.

Representation of Discrete-Time Signals

• Discrete-time signals are defined only at discrete time instants.
• The amplitude between two time instants isn't defined.
• Discrete-time signals are represented by x(n).
• Four ways to represent discrete-time signals:
  • Graphical representation
  • Functional representation
  • Tabular representation
  • Sequence representation

Graphical Representation

• Example: Given x(-2) = -3, x(-1) = 2, x(0) = 0, x(1) = 3, x(2) = 1, and x(3) = 2.
• The signal can be graphed with discrete points on a number line with n as the independent variable (horizontal axis) and x(n) as the dependent variable (vertical axis).

Functional Representation

• Represented as an equation.

• Example: x(n) = { -3, n= -2 { 2, n= -1 { 0, n= 0 { 3, n= 1 { 1, n= 2 { 2, n= 3

• Or in alternative form: x(n) = 2ⁿu(n)

Tabular Representation

• Discrete-time signals represented as a table
  • Example: | n | x(n) | |---|---| | 2 | 3 | | 1 | 2 | | 0 | 0 | | 1 | 3 | | 2 | 1 | | 3 | 2 |

Sequence Representation

• A list within curly brackets
• Example: x(n) = {-3, 2, 0, 3, 1, 2}

Time and Frequency Domains

• Time Domain: Represents a signal as it varies over time (graphically or functionally).
  • Useful for observing how signals change and for audio signals and sensor readings.
• Frequency Domain: Represents a signal in terms of its frequency components (spectrum). Identifies the frequency content within ranges.
• signals like sound or sensor readings
• Example: Given signal x(t) == Asin(2𝝿ft + 𝝋) ,
  • A = Amplitude.
  • f = Frequency.
  • 𝝋 = Phase shift

ADC Unit

• Samples analog signal.
• Quantizes sampled signal.
• Encodes quantized levels to a digital signal.

Sampling of Continuous Signals

• An analog signal (continuous time) is represented on a graph where every point has a value.
• Digitizing an infinite number of points is computationally hard.
• Sampling takes discrete points from the signal (with fixed interval T).

Sampling Theorem

• An analog signal can be perfectly reconstructed from its samples fs (sampling rate) at least twice the maximum frequency (max frequency= fmax) in the signal.

• fs ≥ 2 * fmax

• For example:

  • Speech signal with max frequency of 4 kHz needs sampling rate of at least 8 kHz.
  • Audio signal with max frequency of 20 kHz needs sampling rate of at least 40 kHz.