Basic Concepts

In this expression:

• A is the amplitude,
• f is the frequency.
• w = 2pf is the angular frequency

        Notice how these sounds are what a musician would call an octave apart.

        Sinusoidal signals don't need to start at zero at t = 0.  There are other possibilities.  Here are two sinusoidal signals.

These two signals have the same amplitude and frequency, but they are not the same.  The difference in the two signals is in their phase.  Phase is another parameter of sinusoids.  Consider how we might write mathematical expressions for the signals in the plot above.

• For the "red" signal, we can write:
• v_red(t) = 150sin(2p60t) = 150cos(2p60t - p/2)
• This signal has a phase angle of -p/2.
• For the "blue" signal, we can write:
• v_blue(t) = 150sin(2p60t-p/2) = -150cos(2p60t)
• This signal has a phase angle of -p radians.
• In general, any sinusoidal signal can be written as:
• v_signal(t) = Asin(2pft + f), where:
• A = amplitude,
• f = frequency,
• f = phase.

• Any time you use a sinusoidal signal you have to make an arbitrary decision about where the time origin (t = 0) is located.
• If you have just one signal you can often choose the time origin to be the instant when the signal goes through zero.  Then
• If you have more than one signal, you can often choose one of the signals as a reference - with zero phase - and measure phase from that reference.
• In the example above, we chose the red signal as the reference, and the blue signal has a phase of -p/2 radians.
• We have used the sine function here, but we could also have done everything with cosines.

P3.   Here are two signals.  Both signals have the same amplitude.  Determine the amplitude of the two signals.

P4.   Next, determine the frequency of the two signals.

P5.   Now, determine the phase of the "blue" signal assuming that the "red" signal is the reference.  Give your answer in radians.

        Sinusoidal signals aren't always the most interesting kind of signal.  They keep doing the same thing over and over.  However, other signals which contain information can often be thought of as combinations of sinusoidal signals.  That includes periodic signals - which repeat in time but not sinusoidally - and even non-periodic signals.  Even totally random signals are often viewed as having frequency components and that concept is borrowed from concepts that first arise when you consider sinusoidal signals.

        So, even if you don't ever see a sinusoidal signal again, you may well be trying to deal with sinusoidal components.  As you get into the study of signals you'll deal with numerical algorithms - like the FFT - that decompose signals into sinusoidal components.  You'll find many uses for whatever you learn about sinusoids.

        There are many other kinds of signals besides sinusoidal signals.

• Some signals are periodic, but not sinusoidal.  Those signals will have a Fourier Series representation.  That's just a way of representing periodic signals as sums of sinusoids.  Fourier Series are at the root of numerical algorithms like the Fast Fourier Transform - the FFT - and are widely used in the analysis of signals.
• Other signals may not even be periodic.  Those signals are also of interest, and we can look at how those signals look.

Now, difficult as it is to imagine, this signal can be represented as a sum of sinusoidal signals.  While that is a subject for another lesson, you should be motivated to learn everything you can about sinusoidal signals.  The things you don't learn can prevent you from continuing when you get to concepts like representing this signal with a sum of sinusoidal signals.

        Information carrying signals will vary in time.  A constant signal really doesn't convey any information.  However, when you are dealing with time-varying signals things that you know about constant signals may still hold true.  In particular.

• KVL still holds for time-varying signals.  In fact, KVL holds at every instant of time.
• KCL also is true at every instant of time.