PERIODIC ANALOG SIGNALS:

  PERIODIC ANALOG SIGNALS:

In data communications, we commonly use periodic analog signals and nonperiodic digital signals. Periodic analog signals can be classified as simple or composite. A simple periodic analog signal, a sine wave, cannot be decomposed into simpler signals. A composite periodic analog signal is composed of multiple sine waves.

• Sine Wave
•  Wavelength
•  Time and Frequency Domain
•  Composite Signals
•  Bandwidth

Sine Wave:

Figure a sine wave

 

Figure a sine wave

Figure two signals with the same phase and frequency, but different amplitudes

Frequency and period are the inverse of each other

Figure   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 3.1

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).

 

Frequency

• Frequency is the rate of change with respect to time.
  
• Change in a short span of time means high frequency.
  

§  Change over a long span of time means low frequency

v  If a signal does not change at all, its frequency is zero.

v  If a signal changes instantaneously, its frequency is infinite

v  Phase describes the position of the waveform relative to time 0.

Figure 3.5 Three sine waves with the same amplitude and frequency, but different phases

Example 3.3

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 

 

Figure 3.6 Wavelength and period

Figure 3.7 the time-domain and frequency-domain plots of a sine wave

v  A complete sine wave in the time domain can be represented by one single spike in the frequency domain

Example 3.7

The frequency domain is more compact and useful when we are dealing with more than one sine wave. For example, Figure 3.8 shows three sine waves, each with different amplitude and frequency. All can be represented by three spikes in the frequency domain.

Figure 3.8 the time domain and frequency domain of three sine waves

Signals and Communication

• A single-frequency sine wave is not useful in data communications
  
• We need to send a composite signal, 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.

Composite Signals and Periodicity

•      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.

Example 3.4

Figure 3.9 shows a periodic composite signal with frequency f. This type of signal is not typical of those found in data communications. We can consider it to be three alarm systems, each with a different frequency. The analysis of this signal can give us a good understanding of how to decompose signals.

Figure 3.9 A composite periodic signal

Bandwidth and Signal Frequency

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

Figure 3.12 the bandwidth of periodic and nonperiodic composite signals

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 maximum amplitude of 10 V.

Solution

Let f_h be the highest frequency, f_l 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