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See Attached Document Preview: EEN336: Signals&Systems Spring2009 HW7: DueMarch19,2013 Problem1: DeterminetheFourierseriescoef?cientsofthefollowingsignals: 1. x [n] = 2+cos(n=3+=6)+sin(=2) 1 [...

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EEN336: Signals&Systems Spring2009 HW7: DueMarch19,2013 Problem1: DeterminetheFourierseriescoef?cientsofthefollowingsignals: 1. x [n] = 2+cos(n=3+=6)+sin(=2) 1 [ ] ? 1 k 2. x [n] = (1) (n2k) 2 k=1 Problem2: Whattime-domainsignalshavethefollowingFourierseries/transformrepresentations: 1. X [k] = cos(k=3)+cos(k=2+=3) 1 ? 2 2. X (!) = (!k=4) (j!j ) 2 k=2 Problem 3: The FS representation ofx[n] is given byX[k] = sin(k=3). Use Fourier series proper- ties,withoutdeterminingx[n],todeterminethefollowingDTsignals: 1. v[n] = x[n+1]x[n1] (*: convolution) 2. w[n] = sin(n=2)x[n]. Problem 4: Determine the Fourier transform methods to determine the response of the following systemsforthegiveninput: sin(2n=5) 1. h [n] = andx [n] = sin(n=3+=4)+cos(3n=4) 1 1 n sin(n=3)cos(n=2) 2. h [n] = andx [n] = sin(n=3+=4)+cos(3n=4) 2 2 n jnj n 3. h [n] = (1=2) andx [n] = (1) 3 3 1

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Answered Same Day Dec 22, 2021

Solution

Robert answered on Dec 22 2021
119 Votes
Solution

Problem 1: Determine the Fourier series coefficients of the following signals:
Solution
1. Fourier Series of a given signal is given by:
[ ] ∑ [ ]

Signal
[ ] (



) (


)
[ ] (



) (



)
[ ] (



)

( (





) (





))
[ ]


( (


) (


) (


) (


))
Also we know that Fourier Series is given by:
[ ] ∑ [ ]

By comparison with the standard equation we get.
This means to find X[0], we need to put k=0 in the above equation, this will reduce overall equation to a
constant and hence,
X[0]=3
Similarly put k=1 in the above equation we will get
[ ] ∑ [ ]

This means we need to check the coefficient of



in x[n] this will give second coefficient.
X[1]= (


)
√


similarly
X[-1]= (


)
√


2.
Signal is given by:
[ ] ∑
We can clearly see that periodicity of the signal is 4 with
x[0]=1
x[1]=0
x[2]=-1
x[3]=0
Therefore fourier Series is given by:
[ ] ∑ [ ]

[ ] ∑ [ ]

[ ] [ ] (


) [ ] (


) [ ] (


) [ ] (


)
[ ] (


) (


) (


) (


)
[ ] (


) (


)
[ ]
therefore
[ ]
[ ] [ ]
Problem 2: What time-domain signals have the following Fourier...
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