A discrete-time Fourier transform (the DTFT)
Periodic: Discrete-Time Fourier Transform is a periodic function of w on a period of 2 * pi
Symmetry: even symmetric real part and the imaginary part odd symmetry
Example 1:
>> n=-1:3;
>> x=1;5;
>> k=0:500;
>> x=1:5;
>> X=x*(exp(-j*pi/500)).^(n'*k);
>> magX=abs(X);angX=angle(X);
>> realX=real(X);imagX=imag(X);
>> subplot(2,2,1);plot(k/500,magX);grid
>> title('Magnitude Part')
>> subplot(2,2,2);plot(k/500,angX);grid;
>> title('Angle Part')
>> subplot(2,2,3);plot(k/500,realX);grid;
>> title('Real Part')
>> subplot(2,2,4);plot(k/500,imagX);grid;
>> title('Imag Part')
Example 2:
n=0:10;
x=(0.9*exp(j*pi/3)).^n;
k=-200:200;
w=(pi/100)*k;
X=x*(exp(-j*pi/100)).^(n'*k);
magX=abs(X);angX=angle(X);
realX=real(X);imagX=imag(X);
subplot(2,2,1);plot(w/pi,magX);grid
title('Magnitude Part')
subplot(2,2,2);plot(w/pi,angX);grid;
title('Angle Part')
subplot(2,2,3);plot(w/pi,realX);grid;
title('Real Part')
subplot(2,2,4);plot(w/pi,imagX);grid;
title('Imag Part')
2, DTFT properties
Linear
When the shift: shift corresponding to a frequency domain time domain frequency shift
Frequency shift: a frequency domain time domain, the frequency domain phase shift
Conjugate: Conjugate time domain, the frequency domain corresponding to conjugate
Reverse: When the corresponding reverse domain inversion in the frequency domain
Real sequence symmetry:
Convolution: convolution in the time domain, the frequency domain corresponding to the product
Multiplication: the product of the time domain, the frequency domain corresponds to a convolution period
Energy: Parseval theorem, the energy density spectrum
