MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 PROBLEM SET 4 SOLUTION Home Study Exercise -o&e W 3.63 For an Lti system whose frequency response H(w)= 回≤W and which has a continuous-time periodic input signal a(t) with the following Fourier series representation: a(t)=> lesk(m/a), where a is a real number between 0 and 1 How large must W be in order for the output of the system to have at least 90% of the riod of s(t)? Basically, H(w) is an ideal Low Pass Filter(LPF)and we need to find how wide it needs to be, in order to pass at least 90% of its input's average energy per period (i.e. average power) First, let's rewrite the condition above relating the average powers of the input and out put, with Fourier series coefficients ak and bk, respectively P=∑|l2,P Jbrl The required condition, then, would be P≥BP→∑M2≥R∑a2, where R=09(*) k=-0o Then, lets calculate the Fourier series coefficients of the output, bk bk=B(k)=么、(m,1l≤WF=am,H≤W/a0 10.ka>w=10,>W/o�� �� �� �� �� MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 4 Solution Home Study Exercise - O&W 3.63 For an LTI system whose frequency response is: ⎪ 1, |�| ← W H(j�) = 0, |�| > W. and which has a continuous-time periodic input signal x(t) with the following Fourier series representation: x(t) = �|k| ejk(�/4)t , where � is a real number between 0 and 1 k=−� How large must W be in order for the output of the system to have at least 90% of the average energy per period of x(t)? Basically, H(j�) is an ideal Low Pass Filter (LPF) and we need to find how wide it needs to be, in order to pass at least 90% of its input’s average energy per period (i.e. average power). First, let’s rewrite the condition above relating the average powers of the input and out￾put, with Fourier series coefficients ak and bk, respectively: |bk| 2 Px = |ak| 2 , Py = k=−� k=−� The required condition, then,would be: |bk| 2 Py → RPx � → R |ak| 2 , where R=0.9 (⇒) k=−� k=−� Then, let’s calculate the Fourier series coefficients of the output, bk: ⎪ ⎪ ak, |k�0| ← W ak, |k| ← W/�0 � bk = akH(jk�0) � bk = = 0, |k�0| > W 0, |k| > W/�0 1