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Consider a discrete time periodic signal x [n] = sin(πn/5) . Let ak be the complex Fourier series coefficients of x[n]. The coefficients {ak}, are non-zero when k = Bm±1, where m is any integer. The value of B is _______.(Answer up to the nearest integer)
Correct answer is '10'. Can you explain this answer?
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Most Upvoted Answer
Consider a discrete time periodic signal x [n] = sin(πn/5) . Let ak b...
Explanation:
To find the value of B, we need to understand the properties of the Fourier series coefficients and how they are related to the periodic signal x[n].
Fourier Series Representation:
The complex Fourier series representation of a periodic signal x[n] with a period N can be expressed as:
x[n] = Σ(k=-∞ to ∞) ak * exp(j * 2π * k * n / N)
Here, ak represents the complex Fourier series coefficients.
Finding the Fourier Series Coefficients:
To find the Fourier series coefficients ak, we can use the formula:
ak = (1/N) * Σ(n=0 to N-1) x[n] * exp(-j * 2π * k * n / N)
In our case, the periodic signal x[n] is given as x[n] = sin(πn/5). Therefore, we need to substitute this expression into the formula to find the coefficients ak.
Non-Zero Coefficients:
According to the given statement, the coefficients {ak} are non-zero when k = Bm±1, where m is any integer.
Let's consider the case when k = Bm+1. Substituting this value into the formula for ak, we get:
ak = (1/N) * Σ(n=0 to N-1) sin(πn/5) * exp(-j * 2π * Bm * n / N)
Since the given signal x[n] is a sinusoidal signal, the only non-zero terms in the summation will occur when the sinusoidal term inside the summation is equal to 1. In other words, when πn/5 - 2πBm*n/N = π/2.
Simplifying this equation, we get:
n/5 - 2Bmn/N = 1/2
n(1/5 - 2Bm/N) = 1/2
For this equation to hold true, the term inside the parentheses must be equal to 1/2. Therefore, we have:
1/5 - 2Bm/N = 1/2
2Bm/N = 1/5 - 1/2
2Bm/N = 3/10
Bm/N = 3/20
Bm = 3N/20
Since m can be any integer, we can choose m = 20 to eliminate the fraction:
B(20) = 3N/20
B = 3N/20 * 1/20
B = 3N/400
Given that the periodic signal x[n] has a period N = 20 (since sin(x) has a period of 2π), we can substitute this value into the equation to find the value of B:
B = 3(20)/400
B = 3/20
B ≈ 0.15 ≈ 0 (nearest integer)
Therefore, the value of B is approximately 0, which means the coefficients {ak} are non-zero when k = ±1.
Final Answer:
The value of B, rounded to the nearest integer, is 0.
To find the value of B, we need to understand the properties of the Fourier series coefficients and how they are related to the periodic signal x[n].
Fourier Series Representation:
The complex Fourier series representation of a periodic signal x[n] with a period N can be expressed as:
x[n] = Σ(k=-∞ to ∞) ak * exp(j * 2π * k * n / N)
Here, ak represents the complex Fourier series coefficients.
Finding the Fourier Series Coefficients:
To find the Fourier series coefficients ak, we can use the formula:
ak = (1/N) * Σ(n=0 to N-1) x[n] * exp(-j * 2π * k * n / N)
In our case, the periodic signal x[n] is given as x[n] = sin(πn/5). Therefore, we need to substitute this expression into the formula to find the coefficients ak.
Non-Zero Coefficients:
According to the given statement, the coefficients {ak} are non-zero when k = Bm±1, where m is any integer.
Let's consider the case when k = Bm+1. Substituting this value into the formula for ak, we get:
ak = (1/N) * Σ(n=0 to N-1) sin(πn/5) * exp(-j * 2π * Bm * n / N)
Since the given signal x[n] is a sinusoidal signal, the only non-zero terms in the summation will occur when the sinusoidal term inside the summation is equal to 1. In other words, when πn/5 - 2πBm*n/N = π/2.
Simplifying this equation, we get:
n/5 - 2Bmn/N = 1/2
n(1/5 - 2Bm/N) = 1/2
For this equation to hold true, the term inside the parentheses must be equal to 1/2. Therefore, we have:
1/5 - 2Bm/N = 1/2
2Bm/N = 1/5 - 1/2
2Bm/N = 3/10
Bm/N = 3/20
Bm = 3N/20
Since m can be any integer, we can choose m = 20 to eliminate the fraction:
B(20) = 3N/20
B = 3N/20 * 1/20
B = 3N/400
Given that the periodic signal x[n] has a period N = 20 (since sin(x) has a period of 2π), we can substitute this value into the equation to find the value of B:
B = 3(20)/400
B = 3/20
B ≈ 0.15 ≈ 0 (nearest integer)
Therefore, the value of B is approximately 0, which means the coefficients {ak} are non-zero when k = ±1.
Final Answer:
The value of B, rounded to the nearest integer, is 0.
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Community Answer
Consider a discrete time periodic signal x [n] = sin(πn/5) . Let ak b...
Given the discrete time periodic signal,
So, we have the fourier series coefficients
Also, we have the period of the function as
Since, the fourier series coefficients are also periodic, so we have
i.e. the fourier series coefficients are non zero for
k = 10 m ± 1 where m = 0, 1, 2, ……….
Given that the system coefficients are non-zero for
k = Bm ± 1
We get the value
B = 10
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Consider a discrete time periodic signal x [n] = sin(πn/5) . Let ak be the complex Fourier series coefficients of x[n]. The coefficients {ak}, are non-zero when k = Bm±1, where m is any integer. The value of B is _______.(Answer up to the nearest integer)Correct answer is '10'. Can you explain this answer?
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