Data Communication
CSE 2211
Chapter 3: Signals
ANALOG and DIGITAL Data
Data can be analog or digital.
Analog data refers to information that is continuous and take
continuous values.
Digital data refers to information that has discrete states and
take discrete values.
Note
To be transmitted, data must be
transformed to electromagnetic signals.
ANALOG and DIGITAL Signals
Signals can be analog or digital.
Analog signals can have an infinite number of values in a
range.
Digital signals can have only a limited number of values.
Periodic and Nonperiodic signals
Both analog and digital signals can take one of two
forms: periodic or nonperiodic
Periodic signal consists of repeating pattern within a measurable time
frame, called a period. The completion of one full pattern is called a
cycle.
Nonperiodic signal changes without exhibiting a pattern or cycle that
repeats over time.
Note
In data communications, we commonly use periodic
analog signals and nonperiodic digital signals.
PERIODIC ANALOG SIGNALS
Periodic analog signal completes a pattern within a time
period, and repeats that pattern over subsequent
identical periods.
A simple periodic analog signal is a sine wave.
π
S(t)= A sin (2 ft+ φ)
S: is the instantaneous amplitude
A: peak amplitude (absolute value of its highest intensity)
F: frequency
φ : Phase shift
Peak Amplitude: The peak amplitude of a signal is the absolute value of
its highest intensity, proportional to the energy it carries.
For electric signals, peak amplitude is normally measured in volts.
Two signals with the same phase and frequency, but different amplitudes
Period and frequency
Period ( T ) : amount of time in sec. a signal can complete one
cycle ( amount of time signal takes for one repetition). Period is
formally expressed in seconds.
Frequency ( f ): no of cycles ( period s) per sec. Frequency is
formally expressed in Hertz (Hz).
Note
Frequency and period are the inverse of
each other.
Period and frequency: Two signals with the same amplitude and phase,
but different frequencies
Table 3.1 Units of period and frequency
Example 3.1
The power we use at home has a frequency of 60 Hz.
The period of this sine wave can be determined as
follows:
Example 3.2
Express a period of 100 ms in microseconds.
Example 3.3
The period of a signal is 100 ms. What is its frequency in
kilohertz?
Solution
First we change 100 ms to seconds, and then we
calculate the frequency from the period (1 Hz = 10−3
kHz).
Phase: Phase describes the position of the wave form relative
to time 0.
1. A sine wave with a phase of 0° is not shifted.
2. A sine wave with a phase of 90° is shifted to the left by 1/4 cycle.
3. A sine wave with a phase of 180° is shifted to the left by 1/2 cycle.
Phase: Three sine waves with the same amplitude and frequency,
but different phases
Starts at time 0 with
a zero amplitude. The
amplitude increasing.
Starts at time 0 with a
peak amplitude. The
amplitude decreasing.
Starts at time 0 with a
zero amplitude. The
amplitude decreasing.
Note
Phase is measured in degrees or
radians.
Example 3.4
A sine wave is offset 1/6 cycle with respect to time 0.
What is its phase in degrees and radians?
Solution
We know that 1 complete cycle is 360°. Therefore, 1/6
cycle is
Example of Sine Wave
Wavelength and period
The wavelength of a signal refers to the relationship between frequency (or
period) and propagation speed of the wave through a medium.
The wavelength is the distance a signal travels in one period.
Time and Frequency Domains: A sine wave can be represented either in the time
domain or frequency domain.
Time domain plot: shows changes in signal amplitude with respect to time.
Frequency domain plot: shows relationship between signal frequency and
peak amplitude.
A complete sine wave in the time domain can be represented by
one single spike in the frequency domain.
Note
The frequency domain is more compact and useful when we are
dealing with more than one sine wave.
A single-frequency sine wave is not useful in data communication.
We need to send a composite signal, a signal made of many simple
sine waves.
Composite Signals
■ A composite signal is a signal made of many simple
sine waves.
■ According to Fourier analysis, any composite signal is
a combination of simple sine waves with different
frequencies, amplitudes, and phases.
■ If the composite signal is periodic, the decomposition
gives a series of signals with discrete frequencies.
■ If the composite signal is nonperiodic, the
decomposition gives a combination of sine waves with
continuous frequencies.
A composite periodic signal
Decomposition of a composite periodic signal in the time and
frequency domains
The time and frequency domains of a nonperiodic signal
A nonperiodic composite signal can be a signal created by a microphone
or a telephone set.
In this case, the composite signal cannot be periodic because that implies
that we are repeating the same word or words with exactly the same tone.
Bandwidth
The bandwidth of a composite signal is the difference between the
highest and the lowest frequencies contained in that signal.
Example 3.6
If a periodic signal is decomposed into five sine waves
with frequencies of 100, 300, 500, 700, and 900 Hz, what
is its bandwidth? Draw the spectrum, assuming all
components have a maximum amplitude of 10 V.
Solution
Let fh be the highest frequency, fl the lowest frequency,
and B the bandwidth. Then
The spectrum has only five spikes, at 100, 300, 500, 700,
and 900 Hz (see Figure 3.13).
Figure 3.13 The bandwidth for Example 3.6
Example 3.7
A periodic signal has a bandwidth of 20 Hz. The highest
frequency is 60 Hz. What is the lowest frequency? Draw
the spectrum if the signal contains all frequencies of the
same amplitude.
Solution
Let fh be the highest frequency, fl the lowest frequency,
and B the bandwidth. Then
The spectrum contains all integer frequencies. We show
this by a series of spikes (see Figure 3.14).
Figure 3.14 The bandwidth for Example 3.7
Example 3.8
A nonperiodic composite signal has a bandwidth of 200
kHz, with a middle frequency of 140 kHz and peak
amplitude of 20 V. The two extreme frequencies have an
amplitude of 0. Draw the frequency domain of the
signal.
Solution
The lowest frequency must be at 40 kHz and the highest
at 240 kHz. Figure 3.15 shows the frequency domain
and the bandwidth.
Figure 3.15 The bandwidth for Example 3.8
Fourier Analysis
Note
Fourier analysis is a tool that changes a
time domain signal to a frequency
domain signal and vice versa.
Fourier Series
■ Every composite periodic signal can be
represented with a series of sine and cosine
functions.
■ The functions are integral harmonics of the
fundamental frequency “f” of the composite
signal.
■ Using the series we can decompose any
periodic signal into its harmonics.
Fourier Series
Examples of Signals and the
Fourier Series Representation
Fourier Transform
■ Fourier Transform gives the frequency
domain of a nonperiodic time domain
signal.
Example of a Fourier
Transform
Inverse Fourier Transform
Time limited and Band limited Signals
■ A time limited signal is a signal for which the amplitude
of s(t) is nonzero only during a period of time; the
amplitude is zero everywhere else.
or, s(t) = 0 for t > T1 and t < T2
■ A band limited signal is a signal for which the amplitude
of S(f) is nonzero only for a range of frequencies; the
amplitude is zero everywhere else.
or, S(f) = 0 for f > F1 and f < F2
