The Figure below shows a low-pass decimation filter followed by a downsampler (the M symbol) that...

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The Figure below shows a low-pass decimation filter followed by a downsampler (the "M" symbol) that discards all but every Mth filter output sample. In this problem we explore the changes in signal time-domain amplitude y(m) and frequency-domain magnitude caused by decimation. x(n) ↑ f₁ = 1000 Hz Lowpass w(n) nonrecursive FIR filter M f82 For this problem, our assumptions are: • The lowpass filter in the Figure above has a passband gain of unity and passband width of 0 to 250 Hz. ● • The x(n) sequence contains a 100 Hz sinusoidal component whose time-domain peak amplitude is P. • In the frequency domain, the 100 Hz x(n) sinusoid is located exactly on a 4N-point discrete Fourier transform (DFT) bin center and its 4N-point DFT spectral magnitude is K. Finally, we apply exactly 4N samples of w(n) to the M = 4 (a) What is the fs2 sample rate (in Hz) of the y(m) time domain sequence? (b) What is the peak time-domain amplitude of the 100 Hz downsampler. sinusoid in the w(n) sequence? (c) What is the peak time-domain amplitude of the 100 Hz sinusoid in the y(m) sequence? Justify your answer. (d) What is the equation that defines the downsampled y(m) sequence in terms of the w(n) sequence? Hint: Your solution to this part of the problem will take the form y(m) = w(?). The Figure below shows a low-pass decimation filter followed by a downsampler (the "M" symbol) that discards all but every Mth filter output sample. In this problem we explore the changes in signal time-domain amplitude y(m) and frequency-domain magnitude caused by decimation. x(n) ↑ f₁ = 1000 Hz Lowpass w(n) nonrecursive FIR filter M f82 For this problem, our assumptions are: • The lowpass filter in the Figure above has a passband gain of unity and passband width of 0 to 250 Hz. ● • The x(n) sequence contains a 100 Hz sinusoidal component whose time-domain peak amplitude is P. • In the frequency domain, the 100 Hz x(n) sinusoid is located exactly on a 4N-point discrete Fourier transform (DFT) bin center and its 4N-point DFT spectral magnitude is K. Finally, we apply exactly 4N samples of w(n) to the M = 4 (a) What is the fs2 sample rate (in Hz) of the y(m) time domain sequence? (b) What is the peak time-domain amplitude of the 100 Hz downsampler. sinusoid in the w(n) sequence? (c) What is the peak time-domain amplitude of the 100 Hz sinusoid in the y(m) sequence? Justify your answer. (d) What is the equation that defines the downsampled y(m) sequence in terms of the w(n) sequence? Hint: Your solution to this part of the problem will take the form y(m) = w(?).