Transmitting Signals First Meeting

Signal Processing: Sinewaves Sinewave is a fundamental tool in many areas of science, engineering and mathematics A periodically repeating curve whose shape when plotted. The horizontal axis is time, t, The graph shows how the property y varies as time passes. Example: In electrical signal in a wire, y is the voltage in the wire. The shape shown in the graph can be created by rotating a line of length a at a constant speed about a fixed pivot O: Simulation

Sinewave: Not Symetrical Sometimes a sinewave does not oscillate symmetrically about the horizontal axis. In an electrical signal: y represented voltage, A sinusoidal variation of voltage above and below c, rather than above and below 0.

Sinewave: Amplitude The amplitude is the maximum displacement of the sinewave from its mean (average) position. Simulation

Sinewave: Cycle, Frequency and Period Frequency (f) The number of cycles per second, Example: A sinewave with 5 cycles per second is said to have a frequency of 5 Hz (hertz) Cycle The basic shape of the waveform that repeats indefinitely. Period (T) The time taken to complete one cycle T = 1/f The mains electricity supply is sinusoidal, with a frequency of 50 Hz. What is T? 1/50 = 0.02 s One Cycle Simulation

Sinewave: Phase Phase, or, more correctly, phase shift, Is how far a sinewave is shifted along the horizontal axis relative to another sinewave taken as a reference The blue sinewave is shifted 1/4 cycle to the right of the reference sinewave If a sinewave to be generated by the rotating line a, then a sinewave lagging by a quarter of a cycle is generated by a line b at 90 degrees to line a. (Ninety degrees = ¼ cycle) why? because a complete revolution, 360 degrees, corresponds to one complete cycle of a sinewave.) Simulation

Sinewave: Equation y = a sin(2πft + φ) y represents displacement at time t a represents the amplitude f is the frequency and φ is the phase The term (2πft + φ) represents an angle that is growing as time passes. This angle is measured in radians rather than degrees. For the following sinewave, it is clear that the amplitude a has the value 5 volts. The values f and φ are not so obvious. φ is a quarter of a cycle is 90 degrees, or π/2 radians. Since hhe sinewave lags behind the reference sinewave, so φ = – π/2 radians. (radian = 57.3 degrees) The equation for the sinewave is: y = 5 sin(200πt – π/2) volts

Radian 1 radian ≈ 57.3 degrees The circumference of a circle has a length of 2 π r, One complete rotation = 2 π radians. 2 π radians =360 degrees one radian = (360/2 π) degrees.

Fourier's theorem The theorem states that any periodic signal can be thought of as a sum of a number of sinewaves of different amplitudes, frequencies and phases. Adding sinewaves together creates non-sinusoidal waveforms. This process is known as Fourier synthesis. Simulation a+b+c 

Fourier's theorem: Analysis Fourier analysis is the process of analyzing periodic non-sinusoidal waveforms in order to determine their component sinewaves. Fourier analysis generally involves fairly advanced mathematics It is simply a matter of looking at each term in the series of additions and interpreting it as one of the component sinewaves a+b+c 

Next Week Fourier Spectrum Signal Processing (Bandwidth, Digitization) Modulation and Multiplexing Error Detection and Correction