tic;?%beginning?time
clear;?clc;
%%
%====================
M?=?3;?%?there?are?3?channels
N?=?256;
n?=?-N?:?N;
fs?=?100;??%?sample?frequency
dt?=?1/fs;
t?=?n?*?dt;
t1?=?linspace（-N*dtN*dt（（2*N+1）*M））;
mu0?=?-20;???%?the?chirp?frequency
f0?=?5;???%?center?frequency?of?the?first?signal?x0
f1?=?35;???%?center?frequency?of?the?second?signal?x1
f2?=?70;???%?center?frequency?of?the?third?signal?x2
alfa?=?acot（-mu0）;
du=2*pi*abs（sin（alfa））/（dt*（2*N+1））;
u?=?linspace（-pi*abs（sin（alfa））??pi*abs（sin（alfa））??2?*?N?+?1）;
u1?=?linspace（-pi*abs（sin（alfa））??pi*abs（sin（alfa））?（（2*N+1）*M））;
uu?=?linspace（0??2*pi*abs（sin（alfa））??2?*?N?+?1）;
uu1?=?linspace（0??2*pi*abs（sin（alfa））?（（2*N+1）*M））;

%%
x0?=?exp（j?*?0.5?*?mu0?*?t?.^?2??+?j?*?2?*?pi?*?f0?*?t）;
x1?=?exp（j?*?0.5?*?mu0?*?t?.^?2??+?j?*?2?*?pi?*?f1?*?t）;
x2?=?exp（j?*?0.5?*?mu0?*?t?.^?2??+?j?*?2?*?pi?*?f2?*?t）;

%%%%%?the?frft?of?the?original?signal?
xx0?=?dfrft3（x0??alfa??N??N??dt）;
xx1?=?dfrft3（x1??alfa??N??N??dt）;
xx2?=?dfrft3（x2??alfa??N??N??dt）;
%%
max0_1?=?max（abs（xx0））;
max1_1?=?max（abs（xx1））;
max2_1?=?max（abs（xx2））;
%%
%%%%%?to?plot?the?original?signal?and?its?frft?sprctrum
%%%=========?for?the?first?signal
figure（1）;?
subplot（211）;
plot（treal（x0））;
title（‘the?first?original?signal?x0‘）;
ylabel（‘x0（t）‘）;
xlabel（‘t‘）
%?axis（[-N*dt?N*dt?-1?1]）;

subplot（212）;
plot（uabs（fftshift（xx0）））;
title（‘the?frft?of?the?first?original?signal?x0‘）;
ylabel（‘xx0（u）‘）;
xlabel（‘u‘）;
%%%=========?for?the?second?signal
figure（11）;?
subplot（211）;
plot（treal（x1））;
title（‘the?second?original?signal?x1‘）;
ylabel（‘x1（t）‘）;
xlabel（‘t‘）
%?axis（[-N*dt?N*dt?-1?1]）;

subplot（212）;
plot（uabs（fftshift（xx1）））;
title（‘the?frft?of?the?second?original?signal?x1‘）;
ylabel（‘xx1（u）‘）;
xlabel（‘u‘）;
%%%=========?for?the?third?signal
figure（12）;?
subplot（211）;
plot（treal（x2））;
title（‘the?third?original?signal?x2‘）;
ylabel（‘x1（t）‘）;
xlabel（‘t‘）
%?axis（[-N*dt?N*dt?-1?1]）;

subplot（212）;
plot（uabs（fftshift（xx2）））;
title（‘the?frft?of?the?third?original?signal?x2‘）;
ylabel（‘xx1（u）‘）;
xlabel（‘u‘）;
%%
%%%%%%?interpolation?by?M?=?3
%%%
N2?=?floor（M*（2*N+1）-1）/2;

%%%s0?is?the?first?interpolated?signal
s0?=?upsample（?x0??M）;
ss0?=?dfrft3（s0??alfa??N2??N2??dt/M）;?%ss0?is?the?first?frft?of?s0?with?angle?of?alfa
max0_2?=?max（abs（ss0））;

%%%s1?is?the?second?interpolated?signal
s1?=?upsample（?x1??M）;
ss1?=?dfrft3（s1??alfa??N2??N2??dt/M）;?%ss1?is?the?second?frft?of?s0?with?angle?of?alfa
max1_2?=?max（abs（ss1））;

%%%s2?is?the?third?interpolated?signal
s2?=?upsample（?x2??M）;
ss2?=?dfrft3（s2??alfa??N2??N2??dt/M）;?%ss2?is?the?third?frft?of?s0?with?angle?of?alfa
max2_2?=?max（abs（ss2））;

%%
%%%%%%?filtering
h0?=?zeros（1??2?*?N?*?M?+?3）;
h1?=?h0;
h2?=?h0;

%%%========?filtering?for?the?first?interpolated?signal
h0（[2*N+2?:?4*N+2]）?=?M*ones（1?length（[2*N+1?:?4*N+1]））;?%?the?first?filter?
%?ss0?=?fftshift（ss0）;
v0?=?ss0?.*