Example Sheet 1 With Solutions
1. Definetheterm“signal”anddescribethreedifferenttypes/classesofsignals
A signal conveys information about the behaviour or attributes of some phenomenon, e.g. how a stimulus to a sensor changes over time.
Analogue signals, or continuous time signals, are quantities that vary continuously in time. For example the continuous voltage from a microphone.
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Discrete signals, or digital or sampled signals are usually versions of analogue signals that have been sampled at equal instants in time. The process of sampling an analogue signal to form a discrete signal is known as digitisation or analogue to digital conversion. A discrete signal is usually characterised by a sampling frequency and a resolution, or bit depth.
Binary signals are pulse waveforms which exist in only two states, 0 and 1; for example a square wave.
2. Usingadiagramexplainthedifferencebetweenthetermssensorandtransducer.
Sensor – detect a physical quantity and convert the information to electrical signal Transducer – convert energy from one form to another – e.g. pressure into electrical,
Sensors are transducers, but transducers are more than just sensors. They usually have signal conditioning circuits built in.
3. Describe the ways in which sensors can be classified.
The type of signal produced:
• Analog sensors produce a signal that is continuous in both magnitude and temporal (spatial) content. Most physical measurands are analogue.
• Digital sensors produce an output in discrete steps or states. Digital signals are typically more repeatable, reliable and easier to transmit.
The power requirements of the device:
• Passive devices directly generate an electrical signal in response to an external stimuli. The output signal power comes from the stimulus. (Thermocouples Piezoelectric sensors)
• Active sensors require an external power supply or an excitation signal for their operation. The output signal power comes from the power supply. (Thermistors, Chemo-resistors)
The type of the Measurand
EEEE3089 Sensing system and signal processing 2020-2021
• mechanical / thermal / magnetic etc Input/output relationship:
• the order of the relationship – linear, quadratic, logarithmic.
4. Usingadiagramexplainthedifferencebetweenaccuracyandprecision.
Accuracy is how close to the true value the recorded value is, whereas precision is how repeatable those measurements are. This gives 4 different situations covering high or low accuracy and precision as shown below.
5. A material with a surface acoustic wave velocity of 3000 m/s is measured with a SRAS system where the generations pattern has a line period of 12 microns. What is the frequency of the generated surface acoustic wave?
Velocity =frequency x wavelength.
Frequency = velocity / wavelength = 3000 / 12 microns = 250MHz
6. Show by derivation that the convolution of two signals is equal to the multiplication of their spectra. Why is this result so useful?
𝐷𝐹𝑇(𝑥 ⊛ 𝑦) = 𝑋(𝑘)𝑌(𝑘) 𝑁−1
𝐷𝐹𝑇(𝑥 ⊛ 𝑦) = ∑(𝑥 ⊛ 𝑦)𝑒−𝑗2𝜋𝑛𝑘/𝑁 𝑛=0
𝐷𝐹𝑇(𝑥 ⊛ 𝑦) = ∑ ∑ 𝑥(𝑚)𝑦(𝑛 − 𝑚)𝑒−𝑗2𝜋𝑛𝑘/𝑁 𝑛=0 𝑚=0
EEEE3089 Sensing system and signal processing 2020-2021
𝐷𝐹𝑇(𝑥 ⊛ 𝑦) = ∑ 𝑥(𝑚) ∑ 𝑦(𝑛 − 𝑚)𝑒−𝑗2𝜋𝑛𝑘/𝑁 𝑚=0 𝑛=0
𝐷𝐹𝑇(𝑥 ⊛ 𝑦) = ∑ 𝑥(𝑚)𝑒−𝑗2𝜋𝑚𝑘/𝑁 𝑌(𝑘)
𝑚=0 𝐷𝐹𝑇(𝑥 ⊛ 𝑦) = 𝑋(𝑘)𝑌(𝑘)
This is so useful because a FFT and multiply and a corresponding IFFT is often much faster to compute than direct calculation of the convolution.
This is because convolution scales with N2 where as the operations for FFT scale Nlog(N) So we have (N+2Nlog(N)) for FFT based convolution
7. Explain what is meant by ‘time domain correlation’ and how is it calculated, and the result interpreted?
Time domain correlation is used to work out how similar two signals are.
It is calculated by multiplying the two signals and summing the result as one signal is shifted in time relative to the other signal.
This can be expressed by the following equation where g and h are the two signals. ∞
𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑔, h) = ∫ −∞
The peak in the resultant output tell you where the two signals are most similar. The amplitude of the signal tells you how similar they are a high value (if normalised, this is equal to 1) means a perfect match and 0 means they are not similar at all.
8. Using a specific example, explain how an application can place constraints on the choice of sensors.
Sensor specifications cover a wide range of characteristics. Each characteristic will have a different weighting for a specific application. These characteristics include:
– Power usage
– Sensitivity / resolution / accuracy
– Lifetime / reliability
For the example of a mobile device, where power is limited the use a low of power sensors is important. Additionally, The average life of a mobile phone is <5 years and so this means that the sensors life time does not need to be very high. This allows cheap low power sensors to be used.
EEEE3089 Sensing system and signal processing 2020-2021
𝑔(𝜏 + 𝑡)h(𝑡)𝑑𝑡
EEEE3089 Sensing system and signal processing 2020-2021
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