# 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 4.5
A Sine wave completes one cycle in 4 seconds. What is its frequency?
Solution: 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 ### 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 ## Control of Signals

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