As shown in Figure 1, an input signal x(t) composed of two sinusoids with frequencies 100Hz...

    

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As shown in Figure 1, an input signal x(t) composed of two sinusoids with frequencies 100Hz and 2000Hz goes through an LTI system, which is a low pass filter (5th order Butterworth digital filter) with cutoff frequency 500Hz. The spectrum (magnitude of FFT) of the input signal /X(f) is shown in the first figure of Figure 2. The frequency response of the low pass filter /H(f), the Fourier transform of its impulse response h(t), is shown in the second figure. The spectrum /Y(f) / of the output signal y/t) is shown in the third figure. The figure shows that only the sinusoid with frequency 100Hz passes through the system, and the 2000Hz sinusoid gets filtered out. What is the equation relating /X(f), H(f) and / Y(f)? |X(f)| |H(f)| |Y(f)] 0.8 0.6 0.4 0.2 1.2 -3000 - 0.8 0.6 0.4 0.2 0.8 0.6 T 0.4 I 0.2 1 0 -3000 T T T 1 T T -3000 -2000 -2000 -2000 -1000 -1000 -1000 Frequency Content of Input Signal 0 Frequency (Hz) Filter Characteristics 0 Frequency (Hz) Frequency Content of Filtered Output Signal 0 Frequency (Hz) 1000 1000 1000 2000 2000 2000 3000 3000 3000 Figure 2. (1) spectrum of input signal, (2) frequency response of the low pass filter, (3) spectrum of the output signal x(t), X(f)- LTI, LPF h(t), H(f) y(t), H(f) Figure 1. Low pass filter LTI system with input and output As shown in Figure 1, an input signal x(t) composed of two sinusoids with frequencies 100Hz and 2000Hz goes through an LTI system, which is a low pass filter (5th order Butterworth digital filter) with cutoff frequency 500Hz. The spectrum (magnitude of FFT) of the input signal /X(f) is shown in the first figure of Figure 2. The frequency response of the low pass filter /H(f), the Fourier transform of its impulse response h(t), is shown in the second figure. The spectrum /Y(f) / of the output signal y/t) is shown in the third figure. The figure shows that only the sinusoid with frequency 100Hz passes through the system, and the 2000Hz sinusoid gets filtered out. What is the equation relating /X(f), H(f) and / Y(f)? |X(f)| |H(f)| |Y(f)] 0.8 0.6 0.4 0.2 1.2 -3000 - 0.8 0.6 0.4 0.2 0.8 0.6 T 0.4 I 0.2 1 0 -3000 T T T 1 T T -3000 -2000 -2000 -2000 -1000 -1000 -1000 Frequency Content of Input Signal 0 Frequency (Hz) Filter Characteristics 0 Frequency (Hz) Frequency Content of Filtered Output Signal 0 Frequency (Hz) 1000 1000 1000 2000 2000 2000 3000 3000 3000 Figure 2. (1) spectrum of input signal, (2) frequency response of the low pass filter, (3) spectrum of the output signal x(t), X(f)- LTI, LPF h(t), H(f) y(t), H(f) Figure 1. Low pass filter LTI system with input and output