Exam Questions Flashcards
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]
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
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]?
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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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
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
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.
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])
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
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
Draw a block diagram representation of the DSP filter represented by the difference equation:
y[n]=0.8y[n-1]+x[n-1]
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?
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]
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
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 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?
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 - 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
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
Starts at n=3
x[n] is 0 for n < 3
At n=3, u[n]=1
x[3]=3
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
- 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.
Determine the frequency response |H(Ω)| of a first order difference (FOD) filter. Is this form correct?
- 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
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 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’?
