A signal is described by 
Graph the magnitude and phase of its DTFT over the range ?? ? ? < ? .
This can only be done numerically using the DFT to approximate the DTFT. The approximation is
n0 = -9 ;
n1 = 9 ;
n = [n0:n1]' ; % Vector of discrete-time indices for x[n]
N = length(n) ; % Number of points used to represent x[n]
% Compute x[n]
x = [-log(-n(1:9)+1);log(n(10:end)+1)] ;
% Zero-pad the time domain function to increase frequency domain resolution
padFac = 32 ;
Npad = N*padFac ;
nextra = [n1+1:n1+(Npad-N)]' ;
npad = [n;nextra] ;
xpad = [x;zeros(Npad-N,1)] ;
Npad = length(npad) ; % Number of values used to represent x[n]
Xpad = fft(xpad) ; % DFT of x is the approximation to DTFT of x
kpad = [0:Npad-1]' ; % Harmonic numbers in the range 0 through N-1
Xpad = Xpad.*exp(-j*2*pi*kpad*n0/Npad) % Compensate for the fact that x starts at n=-9, not n=0
% Rearrange k and X for a range of frequencies centered at zero
kpad0 = -floor(Npad/2) ;
kpad = kpad0 + [0:Npad-1]' ;
Wpad = 2*pi*kpad/Npad ; % Vector of discrete-time radian frequencies
Xpad = fftshift(Xpad) ; % Rearrange X for frequencies centered at zero
% Graph the results
figure('Position',[20,20,1500,1500],'PaperPosition',[0.5,0.5,10,10]) ;
subplot(3,1,1) ;
ptr = stem(n,x,'k','filled') ;
set(ptr,'LineWidth',2,'MarkerSize',4) ; grid on ;
xlabel('\itn','FontName','Times','FontSize',24) ;
ylabel('x[{\itn}]','FontName','Times','FontSize',24) ;
set(gca,'FontName','Times','FontSize',18) ;
subplot(3,1,2)
ptr = plot(Wpad,abs(Xpad),'k') ;
set(ptr,'LineWidth',2,'MarkerSize',4) ; grid on ;
xlabel('\Omega','FontName','Times','FontSize',24) ;
ylabel('|X({\ite}^{{\itj}\Omega})|','FontName','Times','FontSize',24) ;
set(gca,'FontName','Times','FontSize',18) ;
subplot(3,1,3)
ptr = plot(Wpad,angle(Xpad),'k') ;
set(ptr,'LineWidth',2,'MarkerSize',4) ; grid on ;
xlabel('\Omega','FontName','Times','FontSize',24) ;
ylabel('Phase of X({\ite}^{{\itj}\Omega})','FontName','Times','FontSize',24) ;
set(gca,'FontName','Times','FontSize',18) ;
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