Chapter 2Digital and Analog Signals

Dr. Husam OstaFall 2013/2014

Introduction

Signal is defined as flow of information. Mathematically,

a signal is defined as a function of one independent variable. Continuous-time signals (analog signals):

defined for every value of time. Example: x(t) = cos(π t) Discrete-time signals:

defined only at certain values of time (sampled signal). Continuous-valued signals:

a signal takes on all possible values on a finite or an infinite range. Discrete-valued signals:

the signal takes on values from a finite set of possible values. Digital signal: A discrete-time signal having a set of discrete values.

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Analog-to-Digital Conversion

In order for a signal to be processed digitally. If the signal is in analog form, it is converted to a digital signal by sampling the analog signal at discrete instants in time. Obtaining a discrete-time signal and then by quantizing its values to a set of discrete values. Quantization is based on rounding or truncation.

If the amplitude of a digital signal is quantized into 32 levels, five bits are required to represent signal amplitude: 2^5=32.

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Analog vs. Digital Signals4

In addition to being represented in analog form, information can also be represented in digital form.

In digital signals, a 1 can be encoded as a positive voltage (5V) while a 0 as zero voltage.

Periodic vs. Aperiodic Signals5

Periodic signal: pattern is repeated over time

Aperiodic (non-periodic) signal: pattern is not repeated over time

Periodic analog signals can be classified into simple or composite. A simple periodic analog signal, a sine wave, cannot be decomposed into

simpler signals. A composite periodic analog signal can be composed into multiple sine

waves. Example speech signals.

Digital Sinusoidal Signal

x(n) = A cos(wn+Ф), -∞ < n < ∞

• where n is an integer variable called the sample number,

• A is the amplitude,

• w is the frequency in radians per sample,

• Ф is the phase in radians,

• w = 2пf where f is the frequency. Information embedded in a signal:

Frequency: how many times a cycle of the signal is repeated in one second. Frequency is reciprocal of period.

Phase: describes start position of signal with respect to time 0 (origin) and is measured in degrees.

Amplitude: value of the signal

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Two signals of same phase and frequency with different amplitudes

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Frequency

Frequency is the rate of change with respect to time. If change occurred in a short period of time,

the signal has high frequency. If change occurred over a long period of time,

the signal has low frequency. If a signal does not change at all, its frequency is zero.

The signal is called DC signal. If a signal changes instantaneously, its frequency is infinite.

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Two signals of same phase and amplitude with different frequencies

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Three signals of same amplitude and frequency with different phases

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The time-domain and frequency-domain plots of a sine wave

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Frequency Domain – Fourier Transform

The frequency domain is more compact and useful when we are dealing with more than one sine wave. For example, the next Figure shows three sine waves, each with different amplitude and frequency.

In the time domain, it is sometimes difficult to distinguish between the signals.

In the frequency domain, the three signals are represented as three spikes.

Fourier Transform: a mathematical transform used to transfer a time-domain signal into its frequency-domain representation.

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Frequency-Domain vs. Time-Domain13

Real-life Examples

The signal received at our homes has a frequency of 60 Hz. The period of this sine wave can be determined as follows:

T = 1/f = 1/60 = 0.0166s = 16.6 ms

If the period of a signal is 100 ms, find its frequency in kilohertz?

100 ms = 0.1 s

f = 1/T = 10 Hz = 0.01 KHz

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Speech Signals = Non-Periodic Signals

The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal. Measured in Hertz.

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Bandwidth of Periodic and Aperiodic Signals16

Note: eachfrequency isidentifiable

Note :frequenciesare all overthe place

Example 1

If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, find its bandwidth? Draw the spectrum (range of frequencies), assuming all components have a maximum amplitude of 10V.

Solution Let fh be the highest frequency, fl the lowest frequency, and B the bandwidth. Then

B = fh - fl = 900 – 100 = 800 MHz

The spectrum has only five spikes, at 100, 300, 500, 700, 900 Hz, as shown in next Figure.

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Example 1…18

Example 2

A non-periodic composite signal has a bandwidth of 200 kHz, with a middle (center) 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.

The lowest frequency is at 40 kHz and the highest frequency is at 240 kHz. The below Figure shows the frequency domain plot of the signal.

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Bit Rate

Bit rate: number of transmitted bits per second (bps). Bit length: the distance one bit occupies on the transmission

medium. Bit length = propagation speed x bit duration.

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Baseband Transmission

Digital transmission using wire as a medium typically “consumes” the entire channel and is labeled as baseband signaling.

The LAN is a common example of baseband signaling.

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Transmission Impairments

When a transmitted analog signal is not received correctly, we have signal impairment. Possible causes: Attenuation: loss of energy in overcoming resistance of the

medium.

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Transmission Impairments…

Distortion: signal shape is changed. Since various frequency components of the signal arrive the receiver at different times.

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Transmission Impairments…

Noise: unwanted signal mixed with useful signal. We use term SNR.

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Signal Wavelength

Signal wavelength = propagation speed x period. λ =c/f where c is the phase speed of the wave that depends

on the medium and f is wave’s frequency.For vacuum, c =3x108m/s

The wavelength of a 100 MHz electromagnetic (radio) wave travelling in vacuum is 3 meters.

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Performance Measure

Throughput is a measure of how fast the data is sent through a network. It is measured in bps.

Latency (delay): how long it takes for an entire message to completely arrive at the destination from the time the first bit is sent out from the source until the time the last bit is arrived at the receiver.

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Modulation/Demodulation

Modulation: the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. The carrier wave has a high frequency and therefore facilitates the transmission of the information signal for long distances.

Demodulation is the method of recovering the information signal.

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Data vs. Signals

Data are entities that convey meaningful information such as computer files, audio file, image.

Signals are electromagnetic encoding of data for the purpose of transmitting data over long distances.

Computer networks and data/voice communication systems transmit signals.

Data and signals can be analog or digital.

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Thank You

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