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How To Say What You Want
Describing Signals
What have we learned?
• Any traveling sinusoidal wave may be described by
y = ym sin(kx  wt + f)
• f is the phase constant that determines where the wave
starts; w = 2pf = 2p/T; k = 2p/l; v = l/T = lf = w/k
• Light always reflects with an angle of reflection equal to
the angle of incidence (angles are measured to the normal).
• When light travels into a denser medium from a rarer
medium, it slows down and bends toward the normal.
n1 sin q1 = n2 sin q2
sin qc = n2/n1
NA = n0 sin qm = (n12 - n22)1/2.
What Else Have We Learned?
• Any periodic function of frequency f0 can be expressed as a
sum over frequency of sinusoidal waves having frequencies
equal to nf0, where n is an integer. The sum is called the
Fourier series of the function, and a plot of amplitude
(coefficient of each sin/cos term) vs. frequency is called the
Fourier spectrum of the function.
• Any non-periodic function (so frequency f0 0) can be
expressed as an integral over frequency of sinusoidal waves
having frequencies. The integral is called the Fourier
transform of the function, and a plot of amplitude vs.
frequency is called the Fourier spectrum of the function.
• The Fourier spectrum of a wider pulse will be narrower than
that of a narrow pulse, so it has a smaller bandwidth.
What Exactly Is Bandwidth, and
Why Do We Care?
• A range of frequencies
• Generally found by taking the frequencies with amplitudes more
than half the maximum amplitude (e.g., on a Fourier spectrum)
• Bandwidth for a medium is the range of frequencies which can
pass through that medium with a minimum of separation
• Sampling theory says that a signal transmitting N different
amplitudes per second requires a bandwidth of at least N/2:
B>N/2
• Usually this ideal is not achieved, and the required bandwidth is
larger
– Grant says B approx N
Pulses and Data
• Can represent binary data with pulses in a variety of ways
• 10110 could look like . . .
Notice that the NRZ
takes half the time of
the others for the
same pulse widths
Non-return-to-zero
(NRZ)
Manchester
Coding
Return-to-zero
(RZ)
Bipolar Coding
Do the Before You Start and the
What Kind of Signal Is It? Parts
of the Activity
Distortion
• No physical change is instantaneous
• If change is too slow, won’t have time to rise before needs to fall
• Results in data loss
Sharp edges
Sizeable rise time
Really Distorted
• Since rise is generally exponential, we define “rise time” to
be time from 10% of max value to 90% of max; “fall time”
is time from 90% to 10%
• To be able to resolve data, the rise time and fall time must
be less than 70% of the bit width
Do the Rest of the Activity
Why do we want to modulate signals?
• An antenna produces EM radiation from standing
waves of current; the length of the antenna must be
at least l/4
• For frequencies in the audio range, that antenna
length must be hundreds of kilometers long!
• If you broadcast radio w/o modulation, only one
signal could be sent at a time in any region; e.g,
you’d only have one radio station, and its area
couldn’t overlap any other radio station.
How do we modulate signals?
• Amplitude modulation:
– A signal with a constant carrier frequency is sent
– The original signal becomes the amplitude of the transmitted signal
– Since the transmitted signal is not a simple sine wave, it has a
bandwidth of Fourier components
• Frequency modulation:
– A signal with a constant carrier frequency is sent
– The original signal becomes the change in frequency of the
transmitted signal
– Since the transmitted signal is not a simple sine wave, it has a
bandwidth of Fourier components
– FM is easier to amplify, since only the frequency determines the
signal.
How do we send these signals?
• Radio antenna (AM frequencies around 1000 kHz,
FM frequencies around 100 MHz)
• TV antenna (VHF frequencies are around 100 MHz,
on either side of FM frequencies, UHF frequencies
around 500 MHz)
These are public transmissions, and so the carrier
frequencies are set and regulated
• Coaxial cable
These are private
• Optical waveguides
transmissions, and sent
over range of frequencies
• ISDN
What exactly is a decibel?
• A ratio, often of power
• BUT, in logarithmic form:
• dB = 10 log (P2/P1)
• e.g., if my received signal is 1/10 as big as my transmitted
signal, my “gain” would be
gain dB = 10 log (1/10) = -10
• The minus sign denotes loss, or a second power less than the
initial power
Why do I care about decibels?
• Signal-to-noise ratios are often given in decibels
• You want the signal to be larger than the noise, so the ratio
(in dB) should be positive
• For digital data, we use bit error rate, not signal-to-noise
• Bit error rate is ratio of wrong bits to total bits - it should be
small, whereas SNR should be large
• Bit error rate can be expressed as a plain number, or in
decibels
Before the next class, . . .
• Re-Read Chapter 3-4 of Grant, focusing on
discussion of modes and of different types
of dispersion.
• Start Homework 3, due next Thursday
(posted shortly)
• Do Activity 05 Evaluation by Midnight
Thursday