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CS601 - Data Communication - Lecture Handout 12

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Problems 4.3
A Sine wave has a frequency of 6 Hz. What is its period?
Solution

Problems 4.3

Problems 4.5
A Sine wave completes one cycle in 4 seconds. What is its frequency?
Solution:

Problems 4.5

Another Way to look at Frequency

  • Measurement of the rate of change
  • The rate at which a sine wave moves from its lowest to its highest point is its frequency
  • A 40 Hz signal has half the frequency of a 80 Hz signal, therefore each cycle takes twice as long to complete one cycle I.e. to go from its lowest to its highest
  • Change in a short Time = High Frequency

Two Extremes Frequency

  • What if a signal does not change at all?
  • What if it maintains a constant voltage level the entire time?
    • In such cases , Frequency is going to be zero
  • If a signal does not change, it will never complete any cycles, and frequency is no. of cycles in 1 second so Freq = 0
  • No change at all ⇒
    • Zero frequency
  • Instantaneous changes ⇒
    • Infinite frequency

Phase

  • Phase describes the position of the waveform relative to time zero
  • If we think of the wave as something that can be shifted backward or forward along the time axis
  • Phase describes the amount of that shift
  • It indicates the status of the first cycle
  • Phase is measured in Degrees or Radians
  • 360 degrees – 2 pi Radians
  • A phase shift of 360 degrees correspond to a shift of a complete period
  • A phase shift of 180 degree correspond to a shift of half a period
  • A phase shift of 90 degree correspond to a shift of quarter a period

Phase

Problem 4.7
A sine wave is offset of a cycle with respect to time zero. What is its phase?
Solution

One Cycle = 360 Degrees

Problem 4.7

Control of Signals

  • Signal can be controlled by three attributes:
    • Amplitude
    • Frequency
    • Phase

Control of Signals- Amplitude

 

Control of Signals- Phase

Time and Frequency Domain

  • Time Domain plots show changes in signal amplitude w.r.t Time
  • It is an Amplitude versus Time Plot
  • Phase and Frequency are not explicitly measured on a Time domain plot
  • To show the relationship between amplitude and Frequency, we can use what is called a Frequency Domain Plot

Time and Frequency Domain

Time and Frequency Domain Example

  • Figure compares the time domain (instantaneous amplitude w.r.t Time) and the Frequency domain (Max amplitude w.r.t Frequency)
  • Low Frequency signal in frequency domain corresponds to a signal with longer period in Time domain & vice versa.
  • A signal changing rapidly in Time domain corresponds to High frequency in Frequency domain
  • Figure shows 3 signals with different frequencies and its time and frequency domain presentations

Composite Signals

  • Second type of Analog Signals, that is composed of multiple sine waves
  • So far we have been focused on simple periodic signals or sine waves
  • Many useful sine waves do not change in a single smooth curve b/w minimum and maximum amplitude.
  • They jump, slide , wobble and spikeAs long as as any irregularities are consistent, cycle after cycle, a signal is still Periodic
  • It can be shown that any periodic signal no matter how complex can be decomposed into a collection of sine waves, each having a measurable amplitude, frequency & phase
  • We need FOURIER ANALYSIS to decompose a composite signal into its components

Composite Signals

  • Figure shows a periodic signal decomposed into two sine waves
  • First sine wave (middle one) has a frequency of ‘6’ while the second sine wave has a frequency of ‘0’
  • Adding these two signals point by point results in the top graph
  • Original signal looks like a sine wave that has its time axis shifted downward
  • This shift is because of DC Component or zero frequency component in the signal
  • If you look at the signal in time domain, a single point is there while in frequency domain , two component freq.'s are there

Summary

  • Sine Waves and its Characteristics
  • Control of Signals
  • Time and Frequency Domain
  • Composite Signals

Reading Sections

  • Section 4.4, 4.5 “Data Communications and Networking” 4th Edition by Behrouz A. Forouzan