The form of the FS representation presented in this question, namely x(t) = Σ x[k]e/kwst k=-00 is termed the exponential FS. In this problem, you need to explore several alternative equivalent, ways of expressing the FS representation for real-valued periodic signals (a) Trigonometric form. i. Show that the FS for a real-valued signal x(t) can be written as x(t) = B[0] + [B[k] cos(kwt) + A[k] sin(kwt) k=1 Where B[k] and A[k] are real-valued coefficients.
The form of the FS representation presented in this question, namely x(t) = Σ x[k]e/kwst k=-00 is termed the exponential FS. In this problem, you need to explore several alternative equivalent, ways of expressing the FS representation for real-valued periodic signals (a) Trigonometric form. i. Show that the FS for a real-valued signal x(t) can be written as x(t) = B[0] + [B[k] cos(kwt) + A[k] sin(kwt) k=1 Where B[k] and A[k] are real-valued coefficients.
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![The form of the FS representation presented in this question, namely
x(t) = Σ x[k]e/kwst
k=-00
is termed the exponential FS. In this problem, you need to explore several alternative, yet
equivalent, ways of expressing the FS representation for real-valued periodic signals.
(a) Trigonometric form.
i. Show that the FS for a real-valued signal x(t) can be written as
x(t) = B[0] + [B[k] cos(kw₁t) + A[k] sin(kw¸t)
k=1
Where B[k] and A[k] are real-valued coefficients.
ii.
Express X[k] in terms of B[k] and A[k].
iii.
Use the orthogonality of harmonically related sines and cosines to show that
B[0] = =
x(t)dt,
2
-75²
S
x(t) cos kwotdt
And
2
A[k] =;
-70²
x(t) sink wo tdt.
iv.
Show that A[k] = 0 if x(t) is even and B[k] = 0 if x(t) is odd.
B[k] =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ffa5d11-a4ff-4f7b-b14a-04f79a08fb0d%2F615c0090-b755-4778-8d81-7030fa431263%2Fe4ha0n_processed.png&w=3840&q=75)
Transcribed Image Text:The form of the FS representation presented in this question, namely
x(t) = Σ x[k]e/kwst
k=-00
is termed the exponential FS. In this problem, you need to explore several alternative, yet
equivalent, ways of expressing the FS representation for real-valued periodic signals.
(a) Trigonometric form.
i. Show that the FS for a real-valued signal x(t) can be written as
x(t) = B[0] + [B[k] cos(kw₁t) + A[k] sin(kw¸t)
k=1
Where B[k] and A[k] are real-valued coefficients.
ii.
Express X[k] in terms of B[k] and A[k].
iii.
Use the orthogonality of harmonically related sines and cosines to show that
B[0] = =
x(t)dt,
2
-75²
S
x(t) cos kwotdt
And
2
A[k] =;
-70²
x(t) sink wo tdt.
iv.
Show that A[k] = 0 if x(t) is even and B[k] = 0 if x(t) is odd.
B[k] =
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