Exam Questions Flashcards

1
Q

If we define u[n] to be a unit step function starting at n=0 and δ[n] an impulse located at n=0, sketch the following discrete signals
x[n]=3u[n+2]-3u[n-4]

A

Unit step function u[n+2] starts at n=-2
signal x[n]=0 for n < -2
x[n] = 1 for n>= -2
When n < -2 x[n]=0

Unit step function u[n-4]: is 1 for n>= 4
so when n >= 4 x[n]=0

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2
Q

The signal x[n] = 5+Sin(2πν/20) + 3Sin(2πn/10) + Sin(2πn/5) is convolved with a rectangular window w[n] composed of 10 successive impulses, each of amplitude 1/10, to produce a new output signal y[n]=x[n]*w[n]. What is the exact output signal y[n]?

A

w[n] rectangular window = {1/10, n=0,1,2,3,4,5,6,7,8,9}
y[n]=x[n]*w[n]

w’[n] = (1/10)w[n] where each impulse in w[n] has an amplitude of 1/10
length of w’[n]=10
y[n]=x[n]*w’[n] for first 10 samples & 0 for the remaining

exact output signal y[n]
y[n]=x[n]w’[n] = x[n](1/10)w[n] for n=0,1,2…9
y[n] = 0 for n = 10,11,12…

y[0]=[5 + sin(2π.0/20) + 3sin(2π.0/10) + sin(2π.0/5)] x 1/10 = 0.5
y[1] = [5 + sin(2π.1/20) + 3sin(2π.1/10) + sin(2π.1/5)] x 1/10 = 0.506
y[2] = [5 + sin(4π/20) + 3sin(4π/10) + sin(4π/5)] x 1/10 = 0.512
y[3] = [5 + sin(6π/20) + 3sin(6π/10) + sin(6π/5)] x 1/10 = 0.518
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3
Q

Find the first 9 terms of the impulse response of the following digital filter described by the following difference equations:
y[n]=0.8y[n-1]+x[n-1]
State whether its a notch-, low-, high, or band-pass filter

A

letting x[n]=δ[n]
n=0,1,2,3,4,5,6,7,8
n=0: h[0]=0.8h[-1]+δ[-1]=0
n=1: h[1]=0.8h[0]+δ[0]=0.8 + 1 = 1.8
n=2: h[2]=0.8h[1]+δ[1]=0.8 x 1.8 = 1.44
n=3: h[3]=0.8h[2]+δ[2]=0.8 x 1.44 = 1.152
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rejects high frequency components of signal = low pass filter

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4
Q

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 5 impulses, of unit amplitude, starting at n=0 and ending at n=4.

A

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])

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5
Q

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 5 impulses, of unit amplitude, starting at n=0 and ending at n=4.

Compute the first nine terms of that response for a processor with the difference equation y[n]=y[n-1]-y[n-2]+1.2x[n]

Explain your rationale / discuss your approach to the solution in a step by step fashion

A

x[n]=δ[n]
n=0: y[0]=y[-1]-y[-2]+1.2δ[0] = 1.2
n=1: y[1]=y[0]-y[-1]+1.2δ[1] = 1.2
n=2: y[2]=y[1]-y[0]+1.2δ[2] = 1.2-1.2=0
n=3: y[3]=y[2]-y[1]+1.2δ[3] = 0-1.2 = -1.2
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y[n]=y[n-1]-y[n-2]+1.2x[n]
need x[n] for 0-> 8
s[0]=h[0] == y[0]=y[-1]-y[-2]+1.2δ[0] = 1.2
s[1]=s[0]+h[1] == h[1]=h[0]-h[1]+1.2δ[1] = 1.2-0+0 = 1.2+1.2 =2.4
s[2]=s[1]+h[2] == h[2]=h[1]-h[0]+1.2δ[2] = 1.2-1.2=0+2.4 =2.4
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n 0 1. 2. 3. 4. 5. 6. 7 8
s[n] 1.2 2.4 2.4 1.2 0. 0. 1.2. 2.4. 2.4
s[n-4] 0 0 0 0 1.2 2.4 2.4 1.2 0
y[n]. 1.2. 2.4. 2.4 1.2 1.2 2.4 3.6. 3.6. 2.4

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6
Q

Draw a block diagram representation of the DSP filter represented by the difference equation:
y[n]=0.8y[n-1]+x[n-1]

A
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7
Q

Show using the example of an impulse δ[n] that multiplication by Z^-n0 has the effect of time delaying any sampled data signal by n0 sampling intervals. Then derive an expression for the phase transfer function φ_δ(Ω) of such a delayed unit impulse. What is the slope of the plot of φ_δ(Ω) vs Ω?

A
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8
Q

Find y[n] and Y(Z), where y[n]=h[n]*x[n] and Y(Z)=H(Z).X(Z) for
x[n]=1,2,3,1,-1,1
h[n]=1,1,1
using both time and Z-domain methods.
Show that Y(Z) is indeed the Z-Transform of y[n]

A
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9
Q

Find the poles and zeroes of the transfer function:
H(z)=(z^3-z^2+0.9z-0.9)/(z^3+0.9z^2)
Plot the poles and zeroes of H(z) on the Argand (complex) plane.
Sketch the approximate frequency response over the interval 0<=Ω<= π.
Find the difference equation that describes the action of the corresponding processor

A
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10
Q

Using the smallest possible number of z-plane poles and zeros, design a digital filter with the following parameters:
- complete frequency rejection at Ω=0 radians,
- complete frequency rejection at Ω=π/4 radians,
- a narrow frequency passband at Ω=2π/3 radians with poles at r=0.9,
- No delay in the output y[n]

Find the Z_Transform of the filter H(z) and also the difference equation describing this filter, y[n]=f(x[n])

A
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11
Q

A filter is described by the block diagram below
Find the difference equation for this processor y[n]=f(x[n]). If the input x[n] is an impulse δ[n], find the Z-Transform of the output, i.e. Y(z) for the following initial conditions
y[-1]=y[-2]=0

Does this case represent the Z-transform of the true, natural impulse response of the processor Y(z)=H(z)? If so, why?

A
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12
Q

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

A

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]

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

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13
Q

If we define u[n] to be a unit step function starting at n=0 and δ[n] an impulse located at n=0, sketch the following discrete signals
x[n]=-3u[n-3

A

Starts at n=3
x[n] is 0 for n < 3
At n=3, u[n]=1
x[3]=3

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14
Q

Suppose a processor has a Z-transform with a pole at z=-0.75 and a zero at z=0.75. Sketch the approximate frequency response of the processor

A
  • When a zero is placed at a given point on the z-plane (z=0.75), the frequency response will be 0 at the corresponding point.
  • A pole on the other position will produce a peak at the corresponding frequency point.
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15
Q

Determine the frequency response |H(Ω)| of a first order difference (FOD) filter. Is this form correct?

A
  • Yes it is correct & can verify it by considering its properties
  • Magnitude of frequency response is maximum at Ω=0
    - |H(0)|=sqrt(2) == filter has a DC gain of sqrt(2)
  • Magnitude of frequency response is minimum at Ω=π, |H(π)|=0
    - filter completely attenuates signals at Nyquist frequency
  • Magnitude of frequency response is symmetric about Ω=π/2, |H(π/2)| =1
    - filter does not introduce any phase shift at frequencies around Nyquist frequency.
  • Frequency response has a single 0 at Ω=0 == filter is HPF
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16
Q

What is a “Kaiser” filter and what are the main parameters used to specify it

A
  • 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.

17
Q

What is an ‘equi-ripple’ filter and what are the main parameters used to specify it?

A
  • 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

18
Q

Describe the physical significance of the impulse response of a digital LTI processor.

A
  • 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.
19
Q

What does the acronym LTI represent and what are the main properties of LTI systems

A

Linear Time Invariant
1. Linearity - LTI system is linear if it follows the principle of superposition.
2. Time-invariance - LTI system is time-invariant if its response to a given input signal does not change with time.
3. Homogeneity - LTI system is homogeneous if its response to a given input signal is proportional to the input signal.
4. Additivity - LTI system is additive if its response to a sum of two input signals is equal to the sum of its response to each individual input signal
5. Shift invariance - LTI system is shift-invariant if its response to a given input signal is the same regardless of the starting time of the input signal
6. Time-reversal invariance - LTI system is time-reversal invariant if its response to a given input signal is the same whether the input signal is played forwards or backwards in time.

20
Q

State the convolution theorem and discuss the importance of convolution and deconvolution in DSP systems

A
  • 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.

21
Q
A
22
Q

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

A

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.

23
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24
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25
Q

What are the possible detrimental effects of truncation of h[n] on FIR filter frequency response and how can the non-rectangular ‘windowing’ or apodization techniques help to ameliorate such effects? State and use the Triangular and Von Hann windows to illustrate your answer.

A
  • Loss of stop band attenuation - truncating h[n] can reduce the stop band attenuation of the FIR filter, making it less effective at attenuating certain frequencies.
  • Increased passband ripple - resulting in less smooth frequency response
  • Widening of transition band - resulting in slower roll-off rate + potentially allowing more frequencies to pass through

Triangular window - tapers the ends of the impulse response to 0 in a linear fashion.
- Has a simple & symmetric shape, with the amplitude gradually decreasing towards the edges of the impulse response.

Von Hand window tapers the ends of the impulse response to 0 in a smoother and more gradual manner than Triangular.
- Has a more complex shape with a smoother transition, resulting in better stop band attenuation and narrower transition bands.

By multiplying the truncated impulse response with these windowing functions, and then applying the tapered impulse response to the FIR filter, the detrimental effects of truncation can be mitigated, resulting in improved frequency response.

26
Q

What is Gibbs phenomenon

A
  • Occurs when a signal is approximated by a Fourier series, and there is a discontinuity in the signal.
  • Characterised by a phenomenon where there is a large overshoot in the vicinity of the discontinuity, followed by a series of oscillations that gradually decay in amplitude.
  • Overshoot and oscillations occur even if the number of terms in the Fourier series is increased.
27
Q

Describe in detail the underlying principle of noise reduction by signal sampling and averaging.

A
  • Underlying principle is based on statistical properties of noise
  • When a signal is sampled and averaged, the random noise components are averaged, while the signal components that are correlated across the samples are reinforced. The result is an increase in signal-to-noise-ratio
  • When a noisy signal is sampled and averaged ‘M’ times, the signal power to noise ratio (SNR) is increased by a factor of M and the signal (voltage) to noise ratio by a factor of sqrt(M)
28
Q

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)

A
29
Q

How are SNR power gain and SNR amplitude gain related?

A

Related by a factor of 2 since power pronto square of amplitude

30
Q

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
31
Q

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’?

A