MU 的历年卷，不是 FZU 的
2024-CS425-Summer
Q1
(a) Explain Shannon’s sampling theorem. Use a diagram to illustrate sampling a simple sinusoid with different sampling rate. Given that CD-quality audio employs a sampling rate of 44,100 Hz, discuss how this relates to the characteristics of human hearing.
Leture 2 SHANNON’S SAMPLING THEOREM
- Any signal can be reconstructed exactly from its samples if the highest frequency present in the signal () is no greater than half the sampling frequency (), i.e.,
- The value is also called the Nyquist frequency
Leture 2 
Leture 2 - The typical range of human hearing is from 20 Hz to 20 kHz.
- To accurately capture the highest frequency humans can hear, the sampling rate must be at least 40 kHz (twice the highest frequency).
- 44.1 kHz exceeds the Nyquist rate, providing a margin to ensure all audible frequencies can be captured without aliasing, and with some headroom for filtering and signal processing.
(b) A signal contains frequency components up to . What is the minimum sampling rate required to accurately digitize this signal according to the Nyquist theorem? Describe what could happen if the signal is sampled at 4 kHz instead.
If the signal is sampled at 4 kHz, which is less than the Nyquist rate (10 kHz), the following problems occur:
Leture 2 （会把高频率映射到低频率的位置，例如 3 kHz →1 kHz，产生错误的频率成分）
- Aliasing: Higher frequency components of the signal are incorrectly mapped to lower frequencies within the sampled range, causing distortion.
（超过 2 kHz 的频率信息会全部丢失）
- Loss of Information: Frequencies above 2 kHz cannot be accurately captured, resulting in a loss of the original signal’s details.
(c) Compare the number of quantization levels and signal-to-noise ratios between 8-bit and 16-bit systems. Given the expression, , calculate the difference in signal-to-noise ratio for these two systems. Explain how this relates to dynamic range and perceived volume after digital-to-analog conversion.
- 8-bit:
- 16-bit:
- Difference:
（SNR 与动态范围和感知音量的关系 → 越高的 bit 能提供越精细的量化，减少噪声）
- The 16-bit system’s higher SNR translates to a wider dynamic range, allowing it to accurately reproduce both very quiet and very loud sounds, where subtle details in quiet environment can be perceived without being masked by noise. This significantly enhances the listening experience compared to an 8-bit system.
(d) Even if the Shannon sampling theorem is strictly observed, an anti-aliasing filter is typically used before sampling. Explain why this is necessary.
This is because real-world signals may contain frequencies higher than the Nyquist limit, which, if not filtered out, can cause aliasing even if the sampling rate is sufficient. The filter ensures that only frequencies below the Nyquist frequency are sampled, preventing distortion and ensuring accurate digitization of the signal.
(e) Say whether the following statements are true or false:
- i. Sampling at exactly twice the highest frequency component of a signal always prevents aliasing. ❌
- It is only in ideal condition.
- ii. Nyquist frequency is half the sampling rate. ✔️
- iii. A higher bit depth in digital audio reduces the dynamic range. ❌
- A higher bit depth increases the dynamic range.
- iv. Increase in sampling rate decreases the audio file size. ❌
- It actually increases the audio file size, as more samples are taken per unit of time.
- v. WAV is a lossless audio file format. ✔️
Q2
(a) Given a sampling frequency of 8000Hz and an analog signal that contains sinusoids at 1800Hz, 3300Hz, 4700Hz and 6210Hz, compute the frequencies at which the sinusoids components will appear in the sampled signal.
- and is less than , so they will remain unchanged in the sampled signal
- and is above , so they will be folded back by subtracting
(c) What is the formula to calculate the magnitude and phase of a given complex frequency coefficient? Find the magnitude and phase (in degree) for the following
- (i)
- (ii)
Q3
(a) Discuss the importance of using complex exponentials in the analysis of signals through the Fourier Transform. How do complex exponentials facilitate the representation of both amplitude and phase information?
（复数指数在傅里叶变换中的重要性）
- Complex exponentials allow decomposition of signals into frequency components for comprehensive domain analysis. By expressing sinusoids, they simplify mathematical operations like differentiation and integration.
（表示信号的幅度和相位信息）
- Given the Fourier Transform
- Amplitude:
- Phase:
(b) Provide the first five sine waves of the Fourier series to approximate a square wave signal. The fundamental being
(c) Given the signal and the impulse response , compute the convolution manually, showing each step of your calculation.
Leture 5 Part 3 0 | 1 * 1 = 1 | ㅤ | ㅤ | 1 |
1 | 1 * 3 = 3 | -3 * 1 = -3 | ㅤ | 0 |
2 | 1 * 5 = 5 | -3 * 3 = -9 | 2 * 1 = 2 | -2 |
3 | 1 * (-4) = -4 | -3 * 5 = -15 | 2 * 3 = 6 | -13 |
4 | 1 * 2 = 2 | -3 * (-4) = 12 | 2 * 5 = 10 | 24 |
5 | ㅤ | -3 * 2 = -6 | 2 * (-4) = -8 | -14 |
6 | ㅤ | ㅤ | 2 * 2 = 4 | 4 |
(d) Write a pseudocode for the Fast Fourier Transform algorithm, assuming the length of the input signal is a power of 2. Include brief comments to explain each step.
Q4
(a) What is the difference between the terms pitch and the fundamental frequency when referring to speech production?
Leture 6 Part 2 The fundamental frequency is the actual measurable frequency of the sound.
The pitch is how we perceive fundamental frequency of a sound, whether or not that frequency is actually present in the waveform.
- 音高 → 人耳如何感知声音的频率
- 基频 → 周期性音波的最低频率
(b) What is Pitch, Formants and Fricatives of a speech signal? Explain them with examples.
Leture 6- Pitch is the perceived frequency of a speech sound. It indicates how high or low a sound is.
- Formant is the resonant frequency of the vocal tract that shape the characteristics of vowel sounds.
- Example: The sound /a/ in “cat” has formants F1 around 730 Hz and F2 around 1090 Hz.
- Fricative is noise-like sound produced by forcing air through a narrow constriction in the vocal tract.
- The sound /s/ in “sun”
(d) The resonance frequencies of the vocal tract tube are called the formant frequencies or more simply, formants. The production of these formants can be modelled by a cascade of IIR filters. Illustrate using a diagram how the flexible vocal tract can be described in this manner. Why are IIR filters used in this model?
Leture 7 Part 1 
Because each formant is typically modeled using an IIR resonator filter. This allows both the formant frequency and its bandwidth to be specified, which is essential for accurately representing the resonance properties of the vocal tract.