Lecture 1: Signals and Signal Characterisation

In this lecture, we look at some of the more basic and frequently encountered analogue signals, with particular reference to the sinusoid. Mathematical formulations for these waveforms will be established and two particular parameters of interest will be defined and calculated: the average (or DC) value and the rms (or root mean square) value. We also give some consideration to the idea of a periodic signal being composed of a number of harmonically related sinusoids.

Learning Outcomes:

On completing this lecture, you will be able to:

  • Sketch and form mathematical expressions for common periodic waveforms;
  • Calculate average and rms values;
  • Carry out power calculations based on Fourier series waveform representation.

1.1Some Basic Signals

(i)The Sinewave

For real world analogue signals that are periodic, sine waves play a fundamental role in characterising the mathematical structure of such signals. Indeed, in our consideration of typical speech signals in the last lecture we visually noted tendencies towards the sinusoidal. As shown above, a general sinusoidal signal v is expressed by

where E is the amplitude of the sine wave,  is the phase angle, and  is the angular frequency expressed in rad/s. Normally, we are not all that interested in the phase of a particular signal but rather in the phase difference between two signals. Thus we will most usually consider the phase angle  of the primary signal to be zero.

Note that in the above formulation, the independent parameter is the angle t, where t carries the time dependence, and the period is 2. Most often, however, we wish to directly represent our signal waveforms as functions of time t and the sinewave is thus shown in the following diagram:

Note that the periodic time T is related to angular frequency  by

where f is the (regular) frequency in Hz.

The sinewave is not the only periodic waveform encountered in analogue electronics. Two others of interest are the following:

(ii) The Squarewave

We can formulate the squarewave mathematically as

while the periodic nature is captured by

for integer n.

Essentially the squarewave is a digital signal, a regular sequence of alternating 0’s and 1’s more commonly termed a clock signal.

(iii) The Sawtooth

This is given mathematically by

and again

The voltage is seen to increase linearly until it reaches E at time T and then switches abruptly back to zero.

1.2Average Value

Apart from amplitude, frequency, and phase, there are a number of additional signal parameters of interest. The first of these is the average or DC value and is defined as

We now apply this to the previously specified waveforms of interest.

Sinewave

And noting thatwe get

As expected, the average value of the symmetrical sinewave is zero.

Squarewave

Sawtooth

While both of the latter two waveforms happen to have dc values of E/2, this is not a general result — as will be seen in the sample problems.

1.3Root Mean Square (RMS)

Consider a periodic signal voltage v(t) applied to a resistor R and resulting in a current i(t):

The instantaneous power delivered to R is

The average power is thus given by

The quantity Vrms is a measure of the average power available in the signal v(t) — it needs the R value to specify the actual power delivered to the resistor. V2rms is a measure of signal strength such as is seen, for example, on a mobile phone handset. We now calculate the Vrms for some of the signal waveforms already encountered.

Sinewave

Again noting that and that , we have

or

Squarewave

It is left as an exercise to show that

We also frequently encounter the squarewave with zero DC value:

For this waveform it may readily be shown that

Sawtooth

Each waveform has its own characteristic rms value.

1.4Fourier Series

The relationship between any periodic waveform and the sinewave was fully developed by the French mathematician Jean Fourier. He basically showed that a periodic function could be written as an infinite series of harmonically related sinusoidal functions. There are a number of equivalent ways in which the relationship may be expressed, one of which is as follows:

Let v(t) represent a signal of period T. Then v(t) may be expanded as

a0 denotes the average value or DC component of the waveform while ak denotes the amplitude of the k’th sinusoid and k its phase angle.

The angular frequency of the first, or fundamental, sinusoid is given by

and we note that all subsequent sinusoids have a frequency which is an integral multiple of the fundamental frequency.

The 20 component is called the second harmonic;

The 30 component is the third harmonic, etc.

Our concern is not with the mathematical problem of deriving the Fourier series for any particular periodic signal but rather with working with the Fourier series as a given. In particular, we are interested in how the available power in a signal distributes across the range of frequencies; this is what is known as the spectrum of the signal. For the present we simply note the following Fourier series expansions of waveforms already encountered.

Squarewave

It may be shown that the squarewave of amplitude E can be expressed

We have already determined the DC value of the squarewave to be indeed E/2. This series expansion contains only odd harmonics.

If the DC component is removed from the waveform leaving

then the Fourier series becomes

The effect of adding components to the series and producing the squarewave is illustrated by the following diagram

K=5 (11, 49) denotes the addition of frequency components up to and including 50, ie three sinusoids. Note that the basic outline of the squarewave quickly emerges but even when a large number of terms are included there still remain errors at the points of discontinuity. Basically, it would take an infinite number of sinusoids to precisely re-construct a perfect squarewave.

Sawtooth

Again it may be shown that the Fourier series for the previously specified sawtooth waveform is given by

All harmonic frequencies are present in this series. Again the underlying harmonic structure is illustrated by the following:

This composite contains all sinusoids up to and including 70.

1.5Power Calculations

Electronic circuits tend to be frequency sensitive, ie they may react differently to sinewaves of differing frequency. The Fourier series tool allows us to determine the underlying sinusoidal structure of a given signal and then to calculate how the electronic circuit effects the individual sinusoidal components, particularly with regards to the distribution of available power across these components. The basic idea is as follows:

Consider a periodic signal v(t) for which, for convenience, the DC component is zero, ie

We have already seen how, given a particular waveform, the available signal power can be calculated as the square of Vrms. This must equal the available power in the individual sinusoids:

Example: The human ear can perceive frequencies up to 20kHz (depending on age!). If a zero-DC square wave is played out through a loudspeaker, what fraction of the available signal power is heard by a listener (of normal hearing) if the squarewave (fundamental) frequency is (a) 15kHz, and (b) 5kHz?

As previously noted, a zero-DC squarewave may be expressed

where f0 is the fundamental frequency in Hz.

We know that the total available signal power is given by

(a)If , then the only component actually perceived is the f0 component

since The perceived signal power may be expressed

The fraction of total power that is perceived is thus

(b)If , then the components actually perceived are the f0 and 3f0 components. The perceived signal power is now given

The fraction of the total power that is perceived is

Stated otherwise, 90% of the power in a squarewave is contained in the first two harmonics.

1.6Concluding Remarks

In this lecture we have considered the mathematical representation of a number of frequently encountered analogue signals. Particular attention has been paid to two signal parameters, the DC (or average) value and the rms value, the latter being a measure of available signal power. The very important idea of regarding a periodic signal as being composed of a number of harmonically related sinusoids has in effect introduced the concept of signal spectrum.

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