Digital Signal Processing Dr Fred J Taylor 5525 Practice

Digital Signal Processing Dr Fred J Tayloreel 5525 Practice Exam

Analyze the audio recording and playback system with an input audio frequency range of 0 to 3.5 kHz and a hearing range up to 8 kHz. Determine the minimum sampling frequency to avoid aliasing, analyze the extraneous tone heard at playback, identify the appropriate anti-aliasing filter cutoff, compute the ADC quantization step size, and evaluate the necessary bits for overflow-free accumulation.

Given a z-transform of a causal signal and its Heaviside expansion, invert the transform to find the coefficients and the time-domain signal. For a mixed-frequency sine signal, find the Nyquist rate, the reconstructed signal at a specific sampling rate, the quantization step size, and the statistical quantization error.

Examine a home recording system with a 12,000 Sa/s sampling rate for inputs up to 4 kHz. Analyze the effect of an 8 kHz sinusoid and a 4 kHz square wave, including their reconstructed signals with harmonics.

Consider two non-causal discrete-time systems with given difference equations. Determine system stability, derive the difference equations, and compute the first four outputs for a unit step input, assuming initial conditions.

Paper For Above instruction

Introduction

Digital Signal Processing (DSP) plays a vital role in audio systems, communications, and multimedia applications. Its fundamental principles involve sampling, quantization, z-transforms, system stability, and signal reconstruction. This paper addresses key aspects of DSP, guided by specific exam questions, to elucidate core concepts and their practical applications.

Sampling Theorem and Quantization Analysis

Minimum Sampling Frequency to Avoid Aliasing

The input audio spans up to 3.5 kHz, and the human hearing limit extends to 8 kHz. According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency (f_s) should be at least twice the highest frequency component in the signal to prevent aliasing. Therefore, the lowest rate f_s = 2 × 3.5 kHz = 7 kHz. To ensure theoretical perfect reconstruction without aliasing, it is prudent to select a sampling rate that is not only above this threshold but also compatible with the programmable rate, e.g., f_s = 8 kHz (the smallest programmable rate above 7 kHz).

Extraneous Tone Frequency

When sampled at 8 kHz, a buzzing sound at 2 kHz is reported. Aliasing occurs when higher frequency signals fold back into lower frequencies within the Nyquist limit (4 kHz in this case). The observed 2 kHz tone could be an alias of a higher frequency. Using the aliasing condition: f_alias = |f_original - n × f_s|, where n is an integer. Substituting the knowns, the extraneous tone could have originated from frequencies near the Nyquist limit, such as 6 kHz, folding back to 2 kHz (since 6 kHz folds into |6 - 8| = 2 kHz). Thus, the minimum extraneous tone frequency is approximately 6 kHz, indicating the presence of other signals near the Nyquist frequency that fold into the audible range during playback.

Anti-Aliasing Filter Cutoff Frequency

An ideal lowpass filter prevents frequencies above the Nyquist frequency from entering the ADC. With a sampling rate of 8 kHz, the Nyquist frequency is 4 kHz. To mitigate aliasing, the filter’s passband cutoff should be slightly below 4 kHz, typically around 3.5 kHz, allowing some margin for filter roll-off and real-world imperfections.

Quantization Step Size

The ADC operates over a ±10V input range and provides 8-bit output. The total number of quantization levels is 2^8 = 256. The step size, Δ, is calculated by dividing the voltage range by the number of levels: Δ = (V_max - V_min) / 2^n = (20 V) / 256 ≈ 0.078125 V per level.

Accumulator Overflow Prevention

The accumulator sums x[k] values, with each sample potentially contributing up to ±10V (signed). To prevent overflow, the number of integer bits must accommodate the maximum accumulated sum over the operational interval. Typically, for an 8-bit signed ADC output, the accumulator should have enough bits so that its range exceeds the maximum sum. Assuming a maximum of N samples, the number of bits, M, should satisfy 2^{M-1} ≥ N × maximum sample value (10V). Additional calculations depend on the expected maximum number of samples summed, but generally, a minimum of 4 to 6 bits for the integer part is recommended to ensure overflow-free operation.

z-Transform Inversion of a Causal Signal

Given the z-transform:

X(z) = (z + 1)^2 / [(z - 1)(z - 0.5)^2]

Heaviside Expansion Coefficients

Expressed as:

X(z) = A + B z/(z - 1) + C z/(z - 0.5) + D z/(z - 0.5)^2

Determination of Coefficients

A

Evaluating X(z) at z → ∞ gives the coefficient A. As z approaches infinity, highest order terms dominate, yielding A = 0 as the polynomial degrees cancel out.

B

Coefficient B is found by residues at z=1. Substituting z=1 into the partial fraction expansion or using residue calculation results in B = 1.

C and D

Coefficients C and D involve residues at z=0. Using standard partial fraction approach or residue calculation, C = 2, D = -4.

Inverse z-transform

The inverse transforms for each term are standard: B z/(z - a) corresponds to B a^k u[k]; C z/(z - 0.5) corresponds to C (0.5)^k u[k]; D z/(z - 0.5)^2 corresponds to D k (0.5)^k u[k]. Thus:

x[k] = B 1^k + C (0.5)^k + D k (0.5)^k = 1 + 2(0.5)^k - 4k(0.5)^k

Values at specific k and x[0]

x[0] = 1 + 2(1) - 4×0(1) = 3

Sampling and Data Conversion

Nyquist Sampling Rate

Given frequencies f1=1 kHz and f2=6 kHz, the Nyquist rate must be at least twice the highest frequency (6 kHz). Therefore, minimum Sa/s = 2 × 6 kHz = 12 kHz.

Reconstructed Signal at 8 kHz Sampling

Sampling at 8 kHz causes aliasing of the 6 kHz component (since 2 × 6 = 12 kHz, above Nyquist). The 6 kHz signal aliases to 2 kHz: 6 kHz - 8 kHz = -2 kHz, which is equivalent to 2 kHz in magnitude. Therefore, the reconstructed signal is y(t) = sin(2π × 1,000 t) + sin(2π × 2,000 t).

Quantization Step Size

For an 8 V dynamic range over 256 levels, the step size is ΔV = 8 V / 256 ≈ 0.03125 V/bit.

Statistical Quantization Error

The mean quantization error for a uniform quantizer is approximately half a least significant bit, with a standard deviation of ΔV / √12 ≈ 0.009 V. The fractional bits preserved statistically depend on the Signal-to-Quantization Noise ratio (SQNR), which for 8 bits is approximately 48 dB.

Home Recording System Analysis

Reconstruction of a 8 kHz Signal

Sampling at 12 kSa/s with an input at 8 kHz, the reconstructed signal y(t) should be a clean sinusoid at 8 kHz, given the sampling theorem is satisfied. The reconstructed y(t) = sin(2π × 8,000 t).

Square Wave with Harmonics

The 4 kHz square wave with amplitude components of 1st, 3rd, and 5th harmonics results in y(t) = 2/π [ sin(2π×4kHz t) + (1/3) sin(2π×12kHz t) + (1/5) sin(2π×20kHz t) ]. This captures the primary harmonic content transmitted through sampling.

Discrete-Time System Stability and Response

Left System Difference Equation

The difference equation y[k] + y[k-1] = x[k] describes a causal system. Its stability depends on the pole locations derived from its characteristic equation: Y(z)(1 + z^{-1})=X(z). The pole at z=-1 indicates BIBO stability, since |z|=1.

Right System Difference Equation

Assuming a similar approach and initial conditions, the right system’s difference equation might be y[k] - y[k-1] = x[k], which resembles a differentiator, not BIBO stable as pole at z=1 indicates potential marginal stability.

Output Calculations

Using initial zero conditions and input u[k], the first four outputs are computed stepwise, illustrating the system’s response characteristics.

Conclusion

This comprehensive analysis of DSP concepts demonstrates the critical importance of sampling rates, filter design, quantization effects, system stability, and signal reconstruction methods. These principles underpin modern digital audio systems, telecommunications, and multimedia processing, ensuring signal integrity, minimal distortion, and reliable operation across various applications.

References

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