Lecture 1

1
Lecture 1 Signals in the Time and Frequency
Domains

• 1. Introduction
• 2. Periodic Signals
• 3. Fourier Series Expansion of Periodic Signals
• 4. Spectral Representation of Periodic Signals
• 5. Duty Cycle of a Rectangular Wave
• 6. RMS Voltage and Power Spectrum of Aperiodic
  Signals
• 7. Discrete Fourier Transform (DFT)
• 8. Signal Strength
• 9. Signal Bandwidth
• 10. Conclusion

2
Introduction

• The tasks of communication is to encode
  information as a signal level, transmit this
  signal, then decode the signal at the receiving
  end.
• An analog signal varies continuously with time,
  and has an infinite number of possible signal
  levels.
• A discrete signal changes only once during a
  certain time interval. The signal value during
  this time interval is one sample, and the
  interval length is called the sampling period.
  Each sample has a infinite number of possible
  signal levels.
• A digital signal is discrete, but each sample has
  a finite number of possible signal levels. The
  limited number of levels means that each sample
  transmits a single information. It also means
  that each sample can be represented as digital
  data, a string of ones and zeroes.
• A digital signal is preferred in computer
  communications because computers already store
  and process information digitally.

3

• In an analog signal, noise are added to the
  signal during transmission. The received signal
  will never be identical to the original signal.
• A digital signal will still be subject to
  noise, but the difference between signal levels
  (an 0101 and an 0110) will be sufficiently
  large so that the receiver can always determine
  the original signal level in each sampling
  period. The regenerated, so that the received
  digital data is an exact replica of the original
  digital data.
• Analog to digital conversion reduces the
  amount of information of the signal by
  approximating the analog signal with a digital
  signal. The sampling period and number of levels
  of the digital signal should be selected in order
  to capture as much information of the original
  signal as possible

 
 
   
4
A
A
f
1/T
A
T0
f
1/T
A
T
1/T
f
5

• Analog and digital signals in the time and the
  frequency
• domain.
• The time domain is simply the signal level
  expressed as a function of time.
• The frequency domain is comprised of amplitude
  and a phase for an infinite number of cosine
  functions. These correspond to the superposition
  of an infinite number of sinusoidal waveforms in
  the time domain.
• To convert from the time domain to the frequency
  domain, we take the Fourier transform of the time
  domain representation. 
• The frequency domain representation of a signal
  does not change with time, but the Fourier
  transform for the signal over an infinitely long
  time period would be impossible. Instead, we
  assume that the signal has some finite duration,
  over a sampling interval
• 1. We assume that outside of the sampling
  interval, the signal repeats itself, so that the
  signal is periodic.
• 2. An alternative assumption is that the
  time-domain signal has a zero value outside of
  the sampling interval, so that the signal is
  aperiodic.

6
   

• A periodic signal satisfies the condition
• Period of the signal.
• Aperiodic. signals
• An even function S (t) S (-t)
• Symmetric.
• The phase of the signal.

7
Fourier Series Expansion of Periodic Signals

• a. There are a finite number of discontinuities
  in the period T.
• b. It has a finite average value for the period
  T.
• c. It has a finite number of positive and
  negative maxima in the
• period T.

f0 Fundamental frequency, n f0
Frequency of each term an , bn Fourier series
coefficients
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
Duty Cycle of a Rectangular Wave
The duty cycle
12
ndinteger, An0
13
RMS Voltage Values and Power Spectrum

• The RMS voltage measures the signals power it
  is the square root of the average value of the
  voltage squared, taken over one period of the
  signal

For a sinusoidal signal
14
Aperiodic Signals 

• The Fourier transform of an aperiodic signal can
  be found from integration of the time-domain
  waveform

X( f ) has continuous variation in both the
amplitude spectrum and the phase spectrum.
Because each point in the amplitude spectrum of
an Aperiodic signal is infinitesimally higher in
frequency than the last point, the voltage at any
point in the amplitude spectrum is
infinitesimally small. The output of the
Fourier transform is therefore not voltage, but
the power spectral density of the waveform. This
value is the voltage of a single spectral line
whose power equals the amount of power contained
an a 1 Hz wide frequency band with constant
spectral density.
15
(No Transcript)
16

• Match the fundamental frequency of the signal to
  the sampling interval. All points on the digital
  spectrum analyzer output will be multiples of the
  inverse of the sampling interval. If the
  sampling interval contains an integer number of
  signal periods, the harmonics of the periodic
  signal will appear as sharp peaks on the
  amplitude spectrum.
• A rectangular wave with a frequency of 2000
  Hz must be sampled with a Digital Spectrum
  Analyzer that samples at a rate of 500,000
  samples per second. How many samples should be
  taken if the sampling interval must contain 15
  complete waveforms?

 
17
Signal Strength

• In signal analysis we frequently want to compare
  the power of two signals.
• The decibel (dB) provides a convenient way to
  measure the difference of two power levels.
• It measures the logarithmic power difference
  between two signals.
• If a signal is attenuated by 2 dB in one stage of
  transmission, then is attenuated by 5 dB in the
  next stage of transmission, we the two decibel
  measurements to find the total attenuation of the
  two stages, in this case 7 dB.
• Signal power equals to waveform power DC power

18
Signal Bandwidth

• A baseband signal is any signal which transmits
  data in the form of the amplitude of the signal
  voltage. The bandwidth of a baseband signal is
  measured from zero frequency upwards to fmax .
• A modulated signal transmits data by modifying
  the amplitude, frequency, or phase of a carrier
  signal. The bandwidth of a modulated signal is
  measured from the minimum frequency fmin  (below
  the carrier frequency) upwards to fmax  (above
  the carrier frequency).
• The full bandwidth of a signal is the frequency
  range that includes all spectrum lines of the
  signal.
• The absolute bandwidth (ABW) of a signal is the
  width of the spectrum that contains 98 of the
  signals total power.
• The effective bandwidth (EBW) (bandwidth) of a
  signal is the width of the spectrum that contains
  at least 50 of the signals total power. This
  is the part of the signal whose power is within 3
  dB of the complete signal.

19
(No Transcript)
20
Conclusion

•   A. one-to-one correspondence exists between
  the representation of a given signal in the time
  domain and in the frequency domain.
• B. The character of the power spectrum allows us
  to determine if a signal is aperiodic or periodic
  in the time domain. A periodic signal has a
  discrete frequency spectrum while an aperiodic
  signal has a continuous spectrum.
• C. The amplitude spectrum and phase spectrum
  together allow us to reconstruct the signal in
  the time domain. If only the amplitude spectrum
  or power spectrum is available, it is possible to
  make some conclusions about features of the
  signal in the time domain.
• ? D. The presence of a DC offset means that the
  signal has a constant component, and the entire
  signal is shifted along the voltage axis in the
  time domain. 
• ? I. If all harmonics divisible by a number n
  are missing for a rectangular periodic signal,
  the rectangular signal in the time domain has a
  duty cycle d equal to 1/n .
• F. The power spectrum allows us to determine the
  full bandwidth and the effective bandwidth of the
  signal.
• G. Bandwidth refers to the range of frequencies
  represented in an analog signal. The bandwidth
  of an analog signal determines the maximum
  sampling rate for a digital signal that is
  accurately transmitted via this analog signal.

21
Error Analysis

• A. The absolute error of a measurement is the
  difference between the ideal value xtheory of a
  quantity (the measurement predicted by theory)
  and the experimentally obtained value xmeasured .

Cr Criteria of Accuracy The relative error is
the error of each measurement divided by the
theoretical value of that measurement.
For parts of the experiment where several
measurements are taken at once (a power spectrum
measurement on the spectrum analyzer), you should
compare the error of each measurement against the
maximum theoretical value in that set of
measurements (generally the theoretical amplitude
of the first harmonic).