Review of Fourier Series

1
Lecture 2

• Review of Fourier Series
• Review of Fourier Transforms
• Example
• Spectrum and bandwidth
• Relationship between data rate and bandwidth
• Analog and digital data transmission

2
Spectrum Bandwidth

• Spectrum
• range of frequencies contained in signal
• Examples
• (1) for s(t) sin(2?f1t) 1/3 sin(2p(3f1)t) the
  spectrum is from f1 to 3f1
• (2) for s(t) square wave of time period T
  (1/f) the spectrum is f to ?
• Absolute bandwidth
• width of spectrum
• Examples
• (1) absolute bandwidth is 2f1
• (2) absolute bandwidth is ?

3
Spectrum Bandwidth

• Effective bandwidth
• Often just bandwidth
• band of frequencies containing most of the energy
• DC Component
• Component of zero frequency

4
Power Spectrum

• The function s(t) specifies the signal in terms
  of voltage or current
• The average power of a time limited signal is
  given by
• P (1/t2 - t1) ?t1-gtt2 x(t)2 dt
• For periodic signal the average power in one
  period (T) is given by
• P (1/T) ?0-gtT x(t)2 dt

5
Power Spectrum

• We would like to know the signal power as a
  function of the frequency -- power spectral
  density
• For periodic signals the power spectrum is
  discrete
• Pj power through the first j harmonics
• (1/4) C02 (1/2) ?n1 -gtj cn2
• For aperiodic signal the power spectrum is
  continuous
• P power contained in a band of frequencies f1
  and f2 2 ?f1 -gt f2 S(f) df

6
Power Spectrum

• Half-power bandwidth
• Range of frequencies at which S(f) has dropped to
  half of its maximum value of power
• Pasevals Power Theorem
• The normalized value of the mean power of a
  periodic signal is equal to the sum of squared
  amplitudes of all harmonic components of signal
  s(t).

7
Relationship Between Data Rate and Bandwidth

• Just like the signal has a bandwidth, any
  transmission system has a bandwidth
• the transmission medium accommodates a limited
  band of frequencies
• This limits the data rate that can be carried
• Consider a periodic square wave s(t) of period T
• Using Fourier series the square wave can be
  expressed as
• s(t) (4?/k) ? n odd n1 -gt ? (1/n)sin(2?nf1t)
  
• where f11/T is the period of the signal

8
Relationship Between Data Rate and Bandwidth

• This waveform has infinite number of harmonics
  and hence infinite bandwidth
• The peak amplitude of the kth component is 1/k
  that of the fundamental frequency
• Most of the power will be concentrated in the
  first few harmonics

9
Case 1

• Let us consider that the square wave is very well
  approximated by the first three harmonics. I.e.,
• s(t) 4/p (sin 2pft 1/3 sin 2p 3f t 1/5 sin
  2p 5f t)

10
signal noise
signal
noise
Figure 3.12
11
Signal to Noise Ratio
signal noise
signal
noise
Figure 3.12
12
Analog and Digital Data Transmission

• Data
• Entities that convey meaning
• Signals
• Electric or electromagnetic representations of
  data
• Transmission
• Communication of data by propagation and
  processing of signals

13
Data

• Analog
• Continuous values within some interval
• e.g. sound, video
• Digital
• Discrete values
• e.g. text, integers

14
Analog (Stream) Date

• Voice, video occur in steady stream
• The data can take continuous values

15
Block (Digital) Data

• Examples
• Text files
• Scanned color documents

Figure 3.1
16
Video Data
_at_ 30 frames/sec 10.4 x 106 pixels/sec
_at_ 30 frames/sec 67 x 106 pixels/sec
17
Converting Analog Data to Digital Data

• Sampling
• What should be the sampling rate?
• Nyquist rate Twice the maximum bandwidth of the
  signal
• Digitize the sample
• Quantization error

18
Transmission Medium

• Type
• Guided medium - e.g. twisted pair, optical fiber
• Unguided medium - e.g. air, water, vacuum
• Characteristic
• Simplex
• signal is transmitted in only one direction
• Half-duplex
• both station may transmit but only one at a time
• Full-duplex
• both stations may transmit simultaneously. The
  medium is carrying signals in both directions.

19
Analog and Digital Transmission

• (a) Analog transmission all details must be
  reproduced accurately

• e.g. AM, FM, TV transmission

(b) Digital transmission only discrete levels
need to be reproduced
20
A Generic Long-Distance Link

• Over distance
• Signal is attenuated
• Noise (thermal and electromagnetic) noise gets
  added.
• Repeaters regenerate signal

Figure 3.7
21
An Analog Repeater

• Amplify the signal
• Eliminate distortion using the equilizer
• Different frequency components are
• Attenuated differently
• Delayed differently
• Noise accumulates

22
A Digital Repeater

• Amplifier and equalizer mitigate some of the
  channel distortion
• Goal is not to regenerate the signal but to
  determine the original pulse
• Regenerate a fresh pulse
• Noise does not accumulate

23
Advantages of Digital Transmission

• Digital technology
• Low cost LSI/VLSI technology
• Data integrity
• Longer distances over lower quality lines
• Capacity utilization
• High bandwidth links economical
• High degree of multiplexing easier with digital
  techniques
• Security Privacy
• Encryption
• Integration
• Can treat analog and digital data similarly