FFT size. Determine K so that K is a power of two

I want to to set The computational frequency resolution a the sampling rate f_s divided by the size of FFT so that I can plot the FFT magnitude

Determine K so that K is a power of two such that the Doppler computational frequency resolution is <= 20 Hz. The computational frequency resolution is the sampling rate divided by the FFT sizeFFT size. Plot the magnitude of the Doppler FFT this is my code

I used the K=size(FFT_y) where delta_f_comp=f_s./K;

but the diminssion is not siffecient or the plot doesnt make since

c=3e8;                                 % speed of light m/s
M=50;                                  % 50 pulses
M_samples=M-1;                         % M samples
m_th= 0:1:M_samples;                   % m_th pulses 
v=30;                                  % the radial velocity 30m/s 
% The radar parameters
f_t= 3e9;                               % frequency of 3.0 GHz
tau= 1e-6;                              % a pulse width of 1 microsecond
PRF=4e3;                                % a PRF of 4.0 KHz
f_s=PRF;                                % sampling rate
R_0=40e3;                               % an initial range of R0 = 40 km
A=1;                                    % set A to 1
theta=0;                                % set theta to 0
%
% calculate
%
PRI=1/PRF;                             % find PRI
T = PRI;                               % T = PRI
f_d=(2*v*f_t)/c;                       % Calculate the Doppler frequency Hz
m_calc=R_0/(v*T);                      % calculate m in equation 8.2
t_m= m_calc*T;                         % claculate the transmited times for the m_th pulse
smp_time=1/f_s;                        % sampling time
t_m_2= m_th*T;                         % estimate the transmited times for the m_th pulse
%
%
y_1=A*exp(1i*((2*pi*f_d*t_m)+theta));         % Calculate Equation 8.28 the data y
y_2=A*exp(1i*((2*pi*f_d*t_m_2)+theta));       % Estimate Equation 8.28 the data y
% b) Using subplot, plot in subplot(2,1,1) the real portion of y. In subplot(2,1,2), plot the imaginary portion.
% Let the x axis be the time of each pulse number (number pulses from 0 to M?1), in msec. From the
% plot, estimate the Doppler frequency in Hz. Calculate the Doppler frequency and compare. Discuss.
%
subplot(2,1,1);
plot (m_th,real(y_2));
subplot(2,1,2)
plot(m_th,imag(y_2));
%
% c) Perform Doppler processing on the data by taking a K?point FFT of y. Determine K so that K is a power
% of two such that the Doppler computational frequency resolution is <= 20 Hz. The computational
% frequency resolution is the sampling rate divided by the FFT size. Discuss.
% 
FFT_y= fft(y_2);                        % FFT of y_2
%K=sqrt(size(FFT_y));                                  
%size_K=size(FFT_y);
K=size(FFT_y)
delta_f_comp=f_s./K;

%fd_r<=20                           % the Doppler computational frequency resolution is <= 20 Hz
%fd_r=smp_r/FFT_s                   % The computational frequency resolution is the sampling rate divided by the FFT size.
%
% d) Looking for a peak in the FFT, determine the Doppler frequency and the associated bin number in the
% FFT. Estimate the corresponding radial velocity. Determine the associated error in Doppler frequency
% and radial velocity. Plot the magnitude of the Doppler FFT results in dB (use 20log10) vs Doppler
% frequency. Set the y axis limits to ?40 to 40 dB. Discuss your results.
% Hint: To set axis limits, do this after your figure plotting is done
% ax = axis;
% axis([ax(1:2) -40 40]);
% Hint: To determine Doppler bin frequencies, use fftshift after calculating the fft. Then the Doppler
% frequencies for plotting will be
% Yfreq = -PRF/2 + delta_f_comp*(0 : (K-1));
% where delta_f_comp is the K?point FFT computational frequency resolution.
%
radl_v=(c*f_d)/(2*f_t);                  % Estimate the corresponding radial velocity
plot(delta_f_comp);
plot(real(FFT_y),delta_f_comp);
                                         % Determine the associated error in Doppler frequency and radial velocity
 
% e) Determine a different positive and a different negative radial velocity that will show up in the same
% FFT bin as the results in part (d). Calculate the corresponding radial velocities. Discuss.

Answer 1

[Daniel M]

If you sampling frequency is 4000 Hz, then to get a resolution of 20 Hz you need to take an N-point fft, where N = 4000/20 = 200.

Therefore, it should be

FFT_y = fft(y_2,200)

However, I suggest giving yourself more samples to work with, such as M = 500, if not even more.

Use the second input for the function size to get the value you want. E.g.

K = size(FFT_y,2);

To plot the frequency spectrum properly, you need to create your frequency vector. Then plot against abs(FFT_y). If you do this properly, you will find that the distance between successive data points is 20 Hz.

I suggest going through the first example on the help document for fft to brush up here.

https://www.mathworks.com/help/matlab/ref/fft.html