NEW EXAM QUESTIONS Flashcards
Distinguish clearly between periodic, aperiodic and random signals. What frequency domain techniques are used for the analysis of the three signal types? PERIODIC
- Periodic - repeat themselves exactly after a fixed time interval
- May be synthesised from a superposition of harmonically related sinusoids.
- May be represented by Fourier Series and exhibit line spectra.
Distinguish clearly between periodic, aperiodic and random signals. What frequency domain techniques are used for the analysis of the three signal types? APERIODIC
- Aperiodic - do not repeat themselves exactly after a fixed time interval.
- May contain a continuum of frequencies (may have some degree of predictability - no well-defined period)
- Represented by the integral Fourier transform.
Distinguish clearly between periodic, aperiodic and random signals. What frequency domain techniques are used for the analysis of the three signal types? RANDOM
-Random signals - have no predictable pattern / structure.
- no well defined frequency content (e.g. thermal, white, shot noise)
- Represented by spectral analysis to characterise the power spectral density of the signal
Distinguish between ‘recursive’ and ‘non recursive’ digital filters (i.e. compare and contrast the characteristics of each).
How are they related to FIR and IIR filter classes?
Use one example of each to illustrate your answer RECURSIVE
Recursive - depend on previous output y[n-m], m=1,2,3,… and current/ previous values of the input x[n-m], m=0,1..
- Concomitantly more computationally efficient than FIR filters
- Have a memory element, allowing filter to have a response that depends on its past inputs + outputs.
- Become unstable at particular frequencies if poorly designed.
- Do not generally yield zero/linear phase characteristics
- Have an infinite impulse response (IIR), meaning that their impulse response does not decay to 0 but has an exponential decay
- e.g. y[n] = y[n-1]+x[n]
Distinguish between ‘recursive’ and ‘non recursive’ digital filters (i.e. compare and contrast the characteristics of each).
How are they related to FIR and IIR filter classes?
Use one example of each to illustrate your answer NON-RECURSIVE
Non-recursive - depend only on present and previous inputs
- does not use feedback / have any memory elements.
- are always stable
- Have a finite impulse response (FIR) meaning the impulse response decays to 0 after a finite number of samples.
- e.g. y[n]=(x[n]+x[n-1]+x[n-2]+…+x[n-N+1])/N
Show using the example of an impulse signal δ[n] that a time shift of n0 samples introduces a frequency dependent phase factor exp(-jn0Ω) into a signal’s spectrum.
Determine the frequency response |H(Ω)| of a first order difference (FOD) filter.
What is a “Kaiser” filter and what are the main parameters used to specify it
- Type of FIR filter
- Designed to provide a tradeoff between the sharpness of the filter’s transition band and the amount of ripple in both the passband and stop band.
Main Parameters:
1. transition bandwidth: frequency range over which the filter transitions from passband to stop band.
2. Passband ripple: maximum allowed variation in passband gain, in dB
3. Stopband attenuation - minimum amount of attenuation required in the stop band, in dB
4. Shape parameter - represented as a beta value and can be adjusted to meet specific design requirements.
What is an ‘equi-ripple’ filter and what are the main parameters used to specify it?
- Have side lobes of approximately similar gain, rather than a maximum near the main lobe and deceasing as one moves away (towards higher frequency) from the main lobe
- Has a passband and stopband ripple that are approximately equal.
- Filter’s magnitude response exhibits small, but constant ripples in both passband and stop band, while maintaining a sharp transition between the two.
Main parameters:
1. passband ripple - amount of variations in the gain of the filter in the passband
2. Stopband ripple - “ “ stopband
3. Cutoff frequency - frequency at which the filter transitions from passband to stop band
4. Filter order - number of poles / zeroes in filters transfer functions. Typically determined by the complexity of desired frequency response + design method used
The signal x[n] = 1 + Sin(2np/40) + Sin(2np/20) + Sin(2np/5) is convolved with a rectangular window w[n] composed of 40 successive impulses, each of amplitude 1/40, to produce a new output signal y[n] = x[n]*w[n]. What is the exact expression for the output signal y[n]?
Describe the physical significance of the impulse response of a digital LTI processor.
- Characterises the behaviour of a digital LTI processor in response to a brief input signal.
- Provides information about how the system responds to inputs at all times.
- The shape of the impulse response can reveal the properties of the system, such as its frequency response and time delay.
- Can be used to determine the output of the system to any input signal by convolving the input signal with the impulse response.
- The Fourier transform of the impulse response gives the frequency response of the system.
State the convolution theorem and discuss the importance of convolution and deconvolution in DSP systems
- Convolution theorem states that the Fourier transform of the convolution of 2 signals is equal to the product of the Fourier transform of the 2 signals - i.e. convolution in the time domain corresponds to multiplication in the frequency domain.
Importance of convolution:
- used to apply filters to signals in the time domain. Important in audio and image processing, where often necessary to remove noise
- Used to implement digital signal processing algorithms, such as fast Fourier transforms (FFT), which are used in spectral analysis and signal processing.
Importance of deconvolution:
- used to reverse the effect of a convolution operation. Importance in equalisation, where it is necessary to remove the effects of a filter from a signal.
- Used in system identification, where the impulse response of a system is estimated by deconvolving the output signal from the input signal.
Explain how the impulse and step responses can be used individually to determine the response (y[n]) of any of the above filters to a pulse (x[n]) comprising 4 unit impulses starting at n = 0.
Impulse response = output of filter when it is given an impulse as input.
Step response = output of filter when it is given a step input. - a signal that starts at zero & increases to a constant value.
Impulse response can be used to find the response y[n] to a pulse x[n] by convolving the input response with the input pulse (response y[n] when input is impulse signal x[n]=δ[n])
Step response can be used to find the response y[n] to a pulse x[n] by convolving the step response with the derivative of the input pulse (response y[n] when input is unit step x[n]=u[n])
Find the first 6 terms of h[n] for a processor with difference equation:
y[n]=-0.2y[n-1]+0.48y[n-2]+x[n]
for (i) zero initial conditions
and (ii) y[-1]=-1.25, y[-2] = -0.52083
explain the physical effect of the nonzero initial conditions
to find h[n], set x[n]=δ[n]
(i) zero initial contions:
y[0]=-0.2y[-1]+0.48y[-2] +δ[0]=1
.
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(ii) nonzero initial contions
y[0]=-0.2y[-1]+0.48y[-2]+δ[0]=1.25
y[1]=-0.2y[0]+0.48y[-1] etc
Nonzero initial conditions cause the output of the filter to start at a non-zero value, which can affect the behaviour of the filter for the first few samples
- Overall shape and characteristics remain the same regardless of initial conditions.
Show, assuming a signal to have noise with a normal probability density distribution superimposed on it, that the SNR power gain after averaging ‘N’ signal samples is given by (S/N)out/(S/N)in = 10Log10(N)
State and illustrate Shannon’s sampling theorem using Fourier analysis for the case of a periodically sampled sinusoidal signal. Show graphically how this principle may be extended to any baseband (DC to some cutoff frequency fmax) signal.
A waveform may be reproduced from its sampled form provided that the sampling frequency is greater than twice the highest frequency component of the waveform
A measurement system with cutoff frequency of 100 kHz provides an output voltage with a r.m.s of 250 mV. Noise, with a r.m.s value of 500 mV, is superimposed on the signal. You are required to sketch the main elements of the system and specify the design parameters of a signal sampling/ averaging system that will provide a reproduction of the original signal with an ‘amplitude’ signal-to-noise ratio (SNR) of 20. How does your design avoid ‘aliasing’?
