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signals2

# signals2 - 1 Signals in the time and frequency domains 1.1...

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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) 170-170 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) 170-170 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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signals2 - 1 Signals in the time and frequency domains 1.1...

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