Find the hilbert-transform pair find the average power of

Question

Assignment -5:

Part I: Signal Interpolation

1. Consider a simple single-mode 128-point sequence

x(n) = sin(πn/64) for n = 0, 1, ... 127

The 32-point subsequence corresponding to a lower sampling rate is in the form of

x' (n) = sin(πn/16) for n = 0, 1, ... 31

Use FFT method to interpolate the subsequence x (n) to 128 points. Plot the resultant 128- point sequence.

2. Consider the 64-point sequence

x(n) = 2^(-n/16) for n = 0, 1, ... 63

The 32-point subsequence corresponding to a lower sampling rate is in the form of

x'(n) = 2^(-n/8) for n = 0, 1, ... 31

Use FFT method to interpolate the subsequence x'(n) to 64 points. Plot the resultant 64-point sequence.

3. Analyze the accuracy of the interpolation processes of these two cases.

Part II: Speech Scrambling

The objective of this exercise is to implement a simple digital speech scrambler. Prior to that, you need to visit the site Audacity. This website provides the basic tools for A/D and D/A conversion. The TA will walk through the tools with you during the discussion session. You also need to recruit a member of the class as the partner of the demo. Your tasks include:

4. Formulation of the design of your speech scrambler, with detailed steps from the continuous-time concept to discrete-time realization, (Note that the design of the scrambler is not unique. There are many feasible versions.)

5. Implementation of the speech scrambler, (Note that the same procedure is for both scrambling and descrambling.)

6. Demonstration of speech scrambling, descrambling, and spectral analysis.

Checkout procedure:

Team member 1:

1. Use the microphone of your computer to record the speech signal of a short sentence.
2. Digitize the speech signal (A/D).
3. Display the spectrum of the digitized speech signal.
4. Reconstruct the speech signal for verification and check against possible distortion (D/A).
5. Apply the speech scrambling procedure to the digitized speech signal.
6. Display the spectrum of the scrambled speech signal.
7. Send the scrambled data samples to your teammate.

Team member 2:

1. Display the spectrum of the received speech signal.
2. Apply the descrambling procedure to the received speech signal.
3. Display the spectrum of the descrambled speech signal.
4. Reconstruct and play the descrambled speech signal.
5. Reply to team member's message. (Record, digitize, scramble, and send).

Full team:
1. Repeat this procedure for two-way verbal conversations 5 times.
2. Compile the entire speech communication record in scrambled form.
3. Descramble and play the communication record.

Assignment -6:

1. Bearing-angle estimation

A passive detection system consists of two acoustic receivers. The separation between the receivers is 2.5 meters. The acoustic propagation speed in air is 343.6 m/s. The acoustic data tracks detected by the receivers are given. The sampling rate of the A/D conversion is 48 MHz. Estimate the bearing angle of the acoustic source.

2. Range estimation

The purpose of this exercise is to perform step-frequency radar imaging using FFT for range estimation.

The data set was taken over a section of the walkway pavement in front of the Broida Hall. The ground-penetrating radar imaging unit scanned along a linear path and took data at 200 spatial positions. The spatial spacing between the data-collection positions is 0.0213 m (2.13 cm).

At each data-collection position, the system illuminates the subsurface region of the walkway area with microwaves in the step-frequency mode, stepping through 128 frequencies with a constant increment, from 0.976 GHz to 2.00 GHz. The relative permittivity εr is approximately 6.0.

The data set is in the form of a (200 x 128) array, corresponding to 128 complex-amplitude values (for the 128 frequencies) at 200 receiving positions.

Your assignment is to perform the image reconstruction of the subsurface profile. Your result should be a two-dimensional image plot (magnitude-only). Please note:

(a) The propagation speed needs to be adjusted by the relative permittivity.
(b) The depth profile needs to be scaled by a factor of two for the round-trip propagation.
(c) The spatial scale of your image profile should be approximately the same, in both directions for effective visualization.

Your report should include (a) a detailed description of your approach and signal processing procedures, (b) resultant image, (c) code, and (d) a summary.

Final Exam

Problem 1:

The frame rate of a camera is N frames/sec. A wheel, initially still, starts to rotate. In the recorded video, the wheel changed the direction of rotation at t = t₁, t₂, t₃, ..... What are the frequencies of the rotation at t = t₁, t₂, t₃, ..... Explain why so.

Problem 2:

The Fourier series expansion of a real periodic signal is in the form of

f(t) = α_o + _n=1∑^∞ a_ncos(nω_ot) + _n=1∑^∞ b_nsin(nω_ot)

(a) Find the Hilbert-transform pair.

(b) Find the average power of the Hilbert-transform pair.

(c) Evaluate the inner product of the Hilbert-transform pair.

Problem 3:

(a) Show that the z → -z^-1 frequency transformation coverts a lowpass filter H_L(z) to a highpass filter

H_H(z) = H_L(-z^-1)

(b) The speech scrambler utilizes the modulation sequence [+1, -1, +1, -1, +1, -1, +1, -1,...] for the redistribution of frequency components. The z→-z^-1 is a lowpass-highpass frequency transformation technique. Compare these two methods and identify the similarities and differences.

Problem 4:

In assignment #5, two data sequences were used for the interpolation experiments. Does the interpolation technique work equally well for both sequences? Explain why so.

Problem 5:

In assignment #6, there were two different approaches for time-delay estimation. One is the correlation method, and the other one is the Fourier transform technique. Describe the relationship between these two methods.

Problem 6:

Three formulas are usually utilized to describe bilinear transformation. For the mapping from the lane to z-plane, the formula is in the form of

z = 1+ βs/1+ βs

For the mapping from the z-plane back to the s-plane, the formula is in the form of

s = (1/β) (z-1/z-2)

The mapping between the analog frequency ω and the DTFT frequency θ is in the form of

 

βω = tan(θ/2)

Bilinear transform has been applied to the design of digital filters regularly.

1. How does the value β change the design procedure?
2. Does the value β change the resultant digital filter? Explain why so.

Problem 7:

The design specifications of a digital lowpass Butterworth filter are:

(a) passband frequency, θ_p = ± π/2
(b) maximum passband attenuation α_max = 0.5 dB
(c) stopband frequency θ_s = ± 7π/10, and
(d) minimum stopband attenuation α_min = 20 dB

Part A: Determine:

1. The design specifications of the equivalent analog lowpass Butterworth filter.
2. The order and cutoff frequency of the analog equivalent.
3. The pole locations of the analog equivalent.
4. The transfer function Ha(s) of the analog equivalent.

Part B: Identify:

1. The order and cutoff frequency of the digital filter,
2. Locations of the poles and zeros of the digital filter,
3. The transfer function Hd(z) of the digital filter, Part C:
1. Partition the transfer function into multiple bilinear or biquad segments

H(z) = H₁(z) H₂(z) H₃(z)...,

and display of the frequency response (amplitude only) of each of the segments, H₁(e^jθ), H₂(e^jθ), H₃(e^jθ)..., of the lowpass filter over the interval from -π to π,

2. Display of the overall frequency response (amplitude only) of H(ejθ) of the lowpass filter over the interval from -π to π, and

3. Verify your design by evaluating the frequency response at the passband and stopband frequencies.