ELECTRONICS 1 – EENG16000
Dr Francesco Fornetti
Section 1A
Section 1A – Signals & Their Representation
▪ Signal sources
▪ Signal representation
▪ Time Domain
▪ Frequency Domain
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Signals and their representation (1)
▪ Signal sources
▪ Two equivalent representations, using Thevenin and Norton equivalent circuits
𝑅𝑆
𝑣𝑆 (𝑡)
Thevenin
𝑖𝑆 (𝑡)
𝑅𝑆
Norton
▪ Although the two are equivalent, Thevenin is preferred when 𝑅𝑆 is low and Norton
when 𝑅𝑆 is high
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Signals and their representation (2)
▪ All real sources are non-ideal and are characterised by a finite internal resistance 𝑅𝑆 .
This is inevitable and reduces the maximum amplitude of the signal that may be
delivered to the load.
𝑅𝑆
𝑣𝑆
𝑅𝐿
𝑣𝑂
𝑅𝐿
𝑣𝑂 = 𝑣𝑆
𝑅𝐿 + 𝑅𝑆
(using voltage divider rule)
▪ From the equation above, it is apparent that, in order to maximise the voltage delivered
to the load, 𝑅𝑠 must be small and certainly much smaller than 𝑅𝐿 , 𝑅𝑠 ≪ 𝑅𝐿
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Signals and their representation (3)
▪ Electrical signals may be either periodic or non-periodic
▪ Periodic signals are entirely predictable and may be represented by specific
mathematical expressions. An obvious example is a sine wave which is fully
characterised by its amplitude, frequency and phase.
𝑠 𝑡 = 𝐴 sin(2𝜋𝑓𝑡 + 𝜃)
▪ When our periodic signal is not quite so simple, we can still represent it as a sum of a
number of sinusoidal signals, with different amplitudes, phases and frequencies. This
may be achieved by means of Fourier Series and Fourier transform.
▪ Really?? Even a square wave??
▪ Yes! However, to have a very accurate representation of this type of signal we would
need an INFINITE number of sinusoidal functions!
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Signals and their representation (4)
▪ Question is, do we need perfection? What can we achieve with a finite number of
terms?
Square W ave FS Partial Sum: n = 3 Error = 0.10029
Square W ave FS Partial Sum: n = 1 Error = 0.19025
1.5
1.5
1 TERM
Square W ave FS Partial Sum: n = 5 Error = 0.067914
1.5
2 TERMS
1
1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1
-1
-1
-1.5
-1.5
0
0.5
1
1.5
0
0.5
1
1.5
-1.5
3 TERMS
0
0.5
1
1.5
▪ By summing only three sinusoids we already have something that resembles a
square wave!
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Signals and their representation (5)
▪ If we use 24 terms we get a
pretty good match between
the actual signal and its
Fourier representation!
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Signals and their representation (6)
▪ We may represent our signal by means of its frequency spectrum. This shows the amplitude and
frequency of the sinusoids which we need to add up in order to reconstruct the signal.
▪ The fundamental component 𝑓0 , i.e. the frequency of our square wave, is the largest in amplitude.
All subsequent terms are at frequencies which are multiples of 𝑓0 and are termed harmonics.
Magnitude
▪ Since harmonics get smaller and smaller as frequency increases, a point will come, along the
frequency axis, beyond which harmonics may be neglected.
𝑓0
3𝑓0
5𝑓0
7𝑓0
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9𝑓0
Frequency
8
Signals and their representation (7)
▪ Although the Fourier transform only applies to periodic signals, it is still useful to look at the
frequency content of non periodic ones.
Magnitude
▪ The main difference is that, in this case, the spectrum is a continuous function of frequency,
rather than a discrete set of frequency lines.
Frequency
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Signals and their representation (8)
▪ We may also sample a signal in the time domain i.e. measure its amplitude at specific points
in time and then store the values as binary numbers
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Signals and their representation (9)
▪ If we choose the time-domain sampling approach we must ask ourselves two questions
▪ Is our sampling frequency fast enough to allow us not to lose information? That is to say, are
our sampling points on the time axis close enough to get all the required amplitude
variations?
▪ Is our resolution sufficient? Once we have measured the amplitude of the signal at a specific
point, we must convert it a binary number. The accuracy of this number depends on how
many bits we have to represent it. If the number of bits is too low, two points which are close
in amplitude may be associated to the same numerical value.
▪ More on this later in the course and in many other units!
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