Edit: As I'm writing out my answers, I'm realizing I'd like to check a lot of these questions. So, thank you for your patience if you decide to check any of these answers for me!

*2.4 Consider a continuous time-domain sinewave, whose cyclic frequency is 500 Hz, defined by x(t) = cos[2π(500)t + π/7].*

Write the equation for the discrete x(n) sinewave sequence that results from sampling x(t) at an fs sample rate of 4000 Hz.

Write the equation for the discrete x(n) sinewave sequence that results from sampling x(t) at an fs sample rate of 4000 Hz.

Does the π/7 matter? I got

**x(n) = cos[2π(1/8)n + π/7]**otherwise.

*2.6 Suppose we used the following statement to describe the Nyquist criterion for lowpass sampling: “When sampling a single continuous sinusoid (a single analog tone), we must obtain no fewer than N discrete samples per continuous sinewave cycle.” What is the value of this integer N?*

Nyquist criterion says we need more than twice the frequency, right? So we would need

**> 2 samples per cycle.**

*2.9 Consider a continuous time-domain sinewave defined by x(t) = cos(4000πt) that was sampled to produce the discrete sinewave sequence defined by x(n) = cos(nπ/2). What is the f_s sample rate, measured in Hz, that would result in sequence x(n)?*

Moving 2π out: The original frequency was 2000 Hz. The sample rate was

**8000 Hz**, s.t. x(n) = cos(2π (2000 Hz) (1/8000 sec/sample) t).

*2.10 Consider the two continuous signals defined by a(t) = cos(4000πt) and b(t) = cos(200πt) whose product yields the x(t) signal shown in Figure P2-10. What is the minimum f_s sample rate, measured in Hz, that would result in a sequence x(n) with no aliasing errors (no spectral replication overlap)?*

So from what I gather, the result of multiplying two cosine waves is in line with the product-to-sum formula for cosine x cosine multiplication. That is, we see two waves, one at 4200 Hz, and another at 3800 Hz. So, in order to prevent aliasing, we have to sample at more than twice the highest frequency. So, anything strictly greater than 4200 Hz x 2 =

**8400 Hz**would be fine.

2.11 Consider a discrete time-domain sinewave sequence defined by x(n) = sin(nπ/4) that was obtained by sampling an analog x(t) = sin(2π(f_o)t) sinewave signal whose frequency is f_o Hz. If the sample rate of x(n) is f_s = 160 Hz, what are three possible positive frequency values, measured in Hz, for f_o that would result in sequence x(n)?

2.11 Consider a discrete time-domain sinewave sequence defined by x(n) = sin(nπ/4) that was obtained by sampling an analog x(t) = sin(2π(f_o)t) sinewave signal whose frequency is f_o Hz. If the sample rate of x(n) is f_s = 160 Hz, what are three possible positive frequency values, measured in Hz, for f_o that would result in sequence x(n)?

I used this equation:

\[x(n) = \sin(2πf_ont_s)\]

Sampling at t_s = 1/160, the simplest f_o is 20 Hz. From here, we know this is aliased with all continuous sequences 20 Hz plus or minus multiples of 160 Hz. So, some possible frequency values are

**20 Hz, 180 Hz, and 340 Hz.**

*2.13 Consider the simple analog signal defined by x(t) = sin(2π700t) shown in Figure P2-13. Draw the spectrum of x(n) showing all spectral components, labeling their frequency locations, in the frequency range −2f_s to +2f_s.*

Would it be correct to simply draw spikes at

**700 Hz, 1700 Hz, -300 Hz, and -1300 Hz?**

*2.16 If a person wants to be classified as a soprano in classical opera, she must be able to sing notes in the frequency range of 247 Hz to 1175 Hz. What is the minimum f_s sampling rate allowable for bandpass sampling of the full audio spectrum of a singing soprano?*

Do I even need to do work using the lower bound for bandpass sampling? The idea is to make replications that are closer to 0Hz, but the bandwidth and lower frequency make it impossible to create replications that won't overlap with the original signal. So, I would use the Nyquist criterion: The bandwidth is 948 Hz, so the sampling frequency needs to be

**greater than 1896 Hz.**

*2.18*

Here is a link to a screenshot of the problem. I'm a little confused by what this problem is asking. Is it just asking for the frequency where the sloping line meets 0 dB? In this case, my answer would be

**f_s - B**. Is that ... correct?

Once again thank you so much for helping me out!