Consider the following block diagram as we explore the changes in signal time$-$domain amplitude and frequency$-$domain magnitude caused by decimation:

We also make the following assumptions:

The lowpass filter shown 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 $100Hzx(n)$ sinusoid is located exactly on a $4$ N $-$point discrete Fourier transform $($DFT$)$ bin center $($again$,$ refer back to chapter $3)$ and its $4$ N $-$point DFT spectral magnitude is K $.$

We apply exactly $4$ N samples of $w(n)$ to the $M=4$ downsampler

a$)$ What is the sample rate $($in Hz $)$ of the $)$ time$-$domain sequence?

b$)$ What is the peak time$-$domain amplitude of the $100$ Hz sinusoid in the $w(n)$ sequence?

c$)$ What is the peak time$-$domain amplitude of the $100$ Hz sinusoid in the $y(m)$ sequence?

d$)$ What is the magnitude of the $100$ Hz spectral component in an N$-$point DFT of $y(m)?$

e$)$ What is the equation that defines the downsampled $y(m)$ sequence in terms of $w(n)?$