Questions & explanations
1. Give an example of a degraded broadcast channel and its capacity region shape.
Consider a binary symmetric broadcast channel where the strong user gets output Y2 = X with no error, and the weak user gets Y1 = X with error probability p. This is degraded because Y1 is a noisier version of Y2. The capacity region is R2 ≤ 1 (since no error), and R1 ≤ 1 - H(p) (the capacity of a BSC with error p). Also, R1+R2 ≤ 1 + (1-H(p))? Actually, the region is R2 ≤ 1, R1 ≤ 1-H(p), and R1+R2 ≤ 1 + (1-H(p))? Wait, correct: The capacity region for this degraded BC is R2 ≤ 1, R1 ≤ 1-H(p), and R1+R2 ≤ 1 + (1-H(p))? No, that's not right. Actually, the region is R2 ≤ 1, R1 ≤ 1-H(p), and R1+R2 ≤ 1 + (1-H(p))? I think it's R2 ≤ 1, R1 ≤ 1-H(p), and there is no sum constraint because the strong user can get both. Actually, the region is R2 ≤ 1, R1 ≤ 1-H(p), and R1+R2 ≤ 1 + (1-H(p))? That would be 2-H(p). But let's not confuse. The typical shape is a rectangle with a sloping side? Actually, it's a pentagon? For degraded BC, the capacity region is R2 ≤ C2, R1 ≤ C1, and R1+R2 ≤ C2 + C1? No, that's MAC. For BC, it's R2 ≤ C2, R1 ≤ C1, and R1+R2 ≤ C2 + C1? That would be the same as MAC? No. Le
2. Give an example of a code that is uniquely decodable but does NOT have the prefix property.
Code: A='0', B='01', C='011'. This is uniquely decodable because the only way to decode is to take the longest match? Actually, it is not uniquely decodable: '011' could be A then C? Wait, A='0', C='011' gives '0'+'011' = '0011' not '011'. Let's use a known example: A='0', B='10', C='010'. This is uniquely decodable? The string '010' could be A then B (0+10) or C (010). So not uniquely decodable. A classic example: A='0', B='01', C='10'. Check '010': could be A,B (0,10) or A,C? Actually C='10' so A then C gives 0+10='010' and also B then? B='01' then? B then? No. Actually, '010' can be A then B (0,01) gives '001' not '010'. Let's use a known uniquely decodable but not prefix: code A='0', B='01', C='011'. The string '011' can be A then C (0,011) gives '0011' no. Actually, it's known that every uniquely decodable code can be transformed into a prefix code. So there is no uniquely decodable code that is not prefix? Wait, there are examples like A='0', B='10', C='110', D='111' is prefix. Actually, the statement is that any uniquely decodable code can be replaced by a prefix code with sam
3. Compare the Gaussian BC capacity region to the Gaussian MAC capacity region for the same total power?
The Gaussian BC and MAC have different constraints. In a MAC, the sum rate is limited by the total received power, while in BC, the sum rate is limited by the broadcast nature. For the same total power and noise, the MAC sum capacity is 0.5 log(1+P/N) for a single user, but with two users, the sum rate can be up to 0.5 log(1+P/N1) + 0.5 log(1+P/N2) in some cases? Actually, MAC sum capacity is 0.5 log(1+P/N) if both users have same noise? For BC, the sum rate is 0.5 log(1+P/N2) for the strong user plus something for weak. Generally, MAC can achieve higher sum rates because users can cooperate? No, they cannot cooperate. In MAC, the sum rate is bounded by the capacity of the multiple access channel, which is typically larger than the BC sum rate for the same power? Not necessarily. For example, with two users and same noise, MAC sum capacity is 0.5 log(1+2P/N) if they use same power? Actually, it's complicated. Better to say: The regions are different shapes; MAC region is a pentagon, BC region is a curve. The sum rate in MAC can be higher because the receiver gets both signals separat
4. If cyclotomic cosets modulo 31 have sizes 1, 5, and 20, what does that imply about the field GF(32)?
Modulo 31, the multiplicative group has order 30. The coset sizes divide 30. Size 1 is {0} only. Size 5 means the element has order dividing 5, so its minimal polynomial has degree 5. Size 20 means degree 20. In GF(32), the multiplicative group has order 31, which is prime, so all non-zero elements have order 31. However, cyclotomic cosets modulo 31 are defined with multiplication by 2 modulo 31. The sizes 5 and 20 indicate that 2 has order 5 modulo 31? Actually, 2^5=32≡1 mod 31, so order is 5. Then cosets have size 5, and the remaining non-zero elements form 6 cosets of size 5? Wait, 30/5=6, so there are 6 cosets of size 5, plus {0}. So no size 20. The statement is inconsistent. Actually, in GF(32), the field has characteristic 2, and cyclotomic cosets modulo 31 are all of size 5 because 2 is primitive? No, 2^5=32≡1 mod 31, so order 5, so cosets size 5. So only sizes 1 and 5 appear.
5. Given a (5,3) regenerating code with parameters (n=5, k=3, d=4, α=2, β=1), explain what each parameter means.
n=5 is the total number of nodes. k=3 means any 3 nodes can recover the original file. d=4 is the number of nodes contacted during repair. α=2 is the storage per node (each node stores 2 units of data). β=1 is the amount of data downloaded from each helper node during repair. So total repair bandwidth is d*β = 4*1 = 4 units. The file size is k*α = 3*2 = 6 units? Actually, file size is typically k*α? No, for an MDS code, file size is k*α? In regenerating codes, the file size is often k*α? Actually, the file size is the total data stored across k nodes, which is k*α, but due to redundancy, the actual file size is less. For an MDS code, the file size is k*α? Wait, need to be careful. In a (5,3) code, the file can be reconstructed from any 3 nodes, so the file size is at most 3*α = 6. But the total storage is 5*α = 10, so overhead 10/6 ≈ 1.67.
6. How can cultivation theory and uses and gratifications give opposite predictions about media effects?
They can give opposite predictions because one focuses on content effects and the other on audience control. For example, cultivation theory predicts that heavy TV viewers will have a more TV-like worldview, regardless of why they watch. But uses and gratifications suggests that if someone watches for a specific reason, like to learn, they might pay more attention and be less influenced by incidental themes. Also, if a viewer is critical and watches for information, they might resist cultivation effects. So, uses and gratifications predicts that active, goal-oriented viewers are less affected, while cultivation predicts that heavy exposure alone matters. Thus, a heavy viewer who watches for learning might show less cultivation than a heavy viewer who watches for escape.
7. Can cultivation theory and agenda-setting ever give opposite predictions? Explain.
Yes, they can give opposite predictions. Agenda-setting says media make issues seem important, but cultivation says media shape beliefs. For example, if media cover a rare disease heavily, agenda-setting predicts people will see it as important. But cultivation predicts that heavy viewers might overestimate their risk of getting it. However, if the media also say the disease is very rare and easily treated, cultivation might not increase fear. So, agenda-setting could make the issue important, but cultivation might not change beliefs if the content is reassuring. Another case: if media focus on a problem but also show it being solved, agenda-setting raises importance, but cultivation may not create a negative worldview. So, the two effects can diverge.
8. Cultivation theory says TV exposure shapes our beliefs. Social cognitive theory says we learn by watching models. How are they similar?
Both theories explain how media influence us through observation. Cultivation theory says that heavy TV viewing leads us to adopt the world as TV shows it, like believing the world is mean. Social cognitive theory says we learn behaviors and attitudes by watching others, including media characters. Both involve learning from media without direct experience. They also both emphasize that repeated exposure matters: cultivation needs long-term viewing, and social cognitive theory says repeated observation strengthens learning. However, cultivation focuses on overall worldview, while social cognitive theory focuses on specific behaviors and the role of rewards and punishments. So, they share the idea that media teach us, but through different mechanisms.
9. How does resonance differ from mainstreaming in terms of the statistical pattern?
Resonance is a statistical interaction where the effect of TV is stronger for people whose real-life experiences match the TV content. For example, TV has a bigger effect on fear of crime for people living in high-crime areas. Mainstreaming, on the other hand, is an interaction where TV reduces differences between groups—heavy viewers from different backgrounds become more alike. So, resonance amplifies existing differences, while mainstreaming diminishes them. In a regression, resonance would show as a positive interaction between TV and a matching real-life variable (like crime rate). Mainstreaming would show as a negative interaction between TV and a demographic variable (like income). Both are interactions but with opposite patterns.
10. Cultivation theory says TV shapes our view of reality. Agenda-setting says media tell us what to think about. How do these two ideas work together?
Cultivation theory and agenda-setting both study media effects but focus on different things. Cultivation theory looks at long-term TV exposure and how it shapes our beliefs about the world, like making us think the world is more dangerous than it is. Agenda-setting says media don't tell us what to think, but what to think about—they highlight certain issues, making them seem important. Together, media can both set the agenda (what we discuss) and cultivate our views on those topics. For example, if news covers crime heavily, it sets the agenda on crime, and heavy TV viewing may cultivate a fear of crime. So, both theories show media influence, but at different levels: agenda-setting on issue salience, cultivation on belief formation.
11. How can third-person effect reduce the impact of cultivation theory?
If people believe that media affect others more than themselves, they might not take steps to protect themselves from media influence. But also, they might dismiss their own cultivation effects because they think they are immune. For example, a heavy TV viewer might say, 'I watch a lot of crime shows, but I know it's not real, so it doesn't affect me.' However, cultivation theory says it still does affect them unconsciously. So, the third-person effect can make people underestimate their own vulnerability, which might mean they don't critically evaluate media content. This could actually strengthen cultivation effects because they don't resist them. So, third-person effect can indirectly allow cultivation to work unchallenged.
12. Explain how social cognitive theory and the elaboration likelihood model (ELM) both address persuasion but in different ways.
Social cognitive theory explains persuasion through observational learning: people adopt attitudes or behaviors after seeing models rewarded or punished. ELM says persuasion happens via two routes: central (careful thinking) and peripheral (cues like attractiveness). Social cognitive theory focuses on learning from others' experiences, while ELM focuses on how people process the message itself. For example, a celebrity endorsement might work through social cognitive theory if the viewer wants to be like the celebrity, and through ELM's peripheral route if the viewer just likes the celebrity. Both theories can be applied together to design effective persuasive messages that provide both a model and strong arguments.