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Unformatted text preview: 1 Signals in the time and frequency domains 1.1 Representing a signal A signal describes how some quantity varies with time. For example, we could write the AC voltage present across a wall outlet as x ( t ) = 170 sin(2 π · 60 t ) . If we were to look at a plot of this, for example on an oscilloscope we would see something like: Time (Seconds) Amplitude (Volts) 170170 1/60 Figure 1: The voltage starts at zero, goes as high as 170 V, as low as 170 V. and back to zero. It makes the transition between these extremes sinusoidally, and this repeats every 1/60 of a second. This interval is called the period T of the signal. If the signal repeats every 1/60 seconds, it must make 60 of such repetitions in a whole second. This number of repetitions per second (Hz) is the frequency f which we can read out of the equation for the function. We can see that T = 1 /f . The above plot is what we call the ‘time domain’ representation of our signal. If we know that we are working with a sine wave, our signal could be completely defined by its amplitude (170 V) and frequency (60 Hz). As a picture, this would be: The arrow 170 Volts high at a frequency of 60 Hz tells us that out signal is a 60 Hz sine wave with a peak voltage of 170 V. This is known as a ‘frequency domain’ representation since the independent variable in the plot is now frequency. Something more interesting occurs when we put a few sine waves together. An example of two waves added together would be y ( t ) = 170 sin(2 π · 60 t ) + 40 sin(2 π · 180 t ) : 1 Frequency (Hz) Amplitude (Volts) 170 60 Figure 2: Time (Seconds) Amplitude (Volts) 170170 1/60 Figure 3: Our original sine wave now has a smaller (amplitude 40), faster (180 Hz) wave added to it. The addition of the new wave causes the regular ripples in the picture and the plot gets a little more complicated. In the frequency domain, this signal gets represented simply be adding an arrow of the appropriate height at 180 Hz: Frequency (Hz) Amplitude (Volts) 170 60 40 180 Figure 4: To see how some more realistic and complex signals can be represented in the frequency domain, we are going to be looking at some sound recordings. A sound, like speech or whistling, is a pressure wave that moves through the air. This wave causes a membrane 2 in our ears to vibrate and our bodies convert this vibration into nerve impulses that are interpreted by our brains. A microphone works in a similar way: the sound waves cause a membrane to vibrate which moves a magnet through a coil according to the patterns of the sound vibrations. The moving magnet makes a voltage in the coil which has the same shape as the sound wave. We can plot this type of waveform as a function of time, just like with out earlier signals....
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 Summer '09
 Osama
 Fourier Series, Frequency, Rxy

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