# Signals and Linear System (PowerPoint) by ert554898

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```									Signals and Linear System

 Fourier Transforms
 Sampling Theorem
Fourier Transform
   Is the extension of Fourier series
to non-periodic signal
   Definition of Fourier transform
   Fourier transform

X ( f )  F[ x(t )]   x(t )e j 2 ft dt


   Inverse Fourier transform     
x(t )  F 1[ X ( f )]   X ( f )e j 2 ft df
                           

   From Fourier series (T0  ) 
1  T
xn       x(t )e j 2 nf t dt x(t )   xn e j 2 nf t
0
0                       0

T0                                 n 
Properties of FT
   For a real signal x(t)
   X(f) is Hermitian Symmetry
   X ( f )  X * ( f )
   Magnitude spectrum is even about the origin (f=0)
   Phase spectrum is odd about the origin

   f, called frequency (units of Hz), is just a parameter of FT
that specifies what frequency we are interested in
looking for in the x(t)
   The FT looks for frequency f in the x(t) over - < t < 
   F(f) can be complex even though x(t) is real
   If x(t) is real, then Hermitian symmetry
Properties of FT
   Linearity
   F[ x1 (t )   x2 (t )]   F[ x1 (t )]   F[ x2 (t )]

   Duality
   If   X ( f )  F[ x(t )]        , Then         F[ X (t )]  x( f )

   Time Shift
   A shift in the time domain results in a phase shift
in the frequency domain
    F ( x(t  t ))  e  X ( f ) ( e  X ( f )  X ( f ) )
0
 j 2 ft0                j 2 ft0
Properties of FT
   Scaling
   An expansion in the time domain results in a
contraction in the frequency domain, and vice
versa
1   f
   F [ x(at )]      X( ) , a  0
a   a
Properties of FT
   Modulation
     Multiplication by an exponential in the time
domain corresponds to a frequency shift in the
frequency domain
     F[e j 2 f0t x(t )]  X ( f  f 0 )
1
F[ x(t ) cos(2 f 0t )]  [ X ( f  f 0 )  X ( f  f 0 )]
2

x(t)

cos
X(f)                                                  -f0           f0
-f0           f0
Properties of FT
   Differentiation
   Differentiation in the time domain corresponds to
multiplication by j2f in the frequency domain
d
   F[    x(t )]  j 2 f  X ( f )
dt
dn
F [ n x(t )]  ( j 2 f ) n  X ( f )
dt
Properties of FT
   Convolution
   Convolution in the time domain is equivalent to
multiplication in the frequency domain, and vice
versa
   F[ x(t )  y(t )]  X ( f )Y ( f )
F[ x(t ) y(t )]  X ( f )  Y ( f )
Properties of FT
   Parseval’s relation
                        
   
x(t ) y* (t )dt   X ( f )Y * ( f )df


   Energy can be evaluated in the frequency
domain instead of the time domain
   Rayleigh’s relation
                                           
       x(t ) x (t )dt          x(t ) dt  
*                       2                    2
                                                        X ( f ) df
                                        
More on FT pairs
   See Table 1.1 at page 20
   Delta function  Flat
   Time / Frequency shift
   Sin / cos input
   Periodic signal  impulses in the frequency domain
   sgn / unit step input
   Rectangular  sinc
   Lambda  sinc2
   Differentiation
   Pulse train with period T0
   Periodic signal  impulses in the frequency domain
FT of periodic signals
   For a periodic signal with period T0
   x(t) can be expressed with FS coefficient

x(t )    xe
n 
n
j 2 nt / T0

   Take FT                                                     
X ( f )  F [ x(t )]  F[  xn e j 2 nt / T0 ]
n 
                                     
n
     
n 
xn F [e   j 2 nt / T0
]   
n 
xn ( f 
T0
)

   FT of periodic signal consists of impulses at
harmonics of the original signal
FS with Truncated signal
   FS coefficient can be expressed using FT
   Define truncated signal
   T0      T0
              ,  t 
 x(t )
xT0 (t )     2       2
0

, otherwise
   FT of  truncated signal: X T0 ( f )  F[ xT0 (t )]
   Expression of FS coefficient
1        n
xn       X T0 ( )
T0       T0
Spectrum of the signal
   Fourier transform of the signal is called the
Spectrum of the signal
   Generally complex
   Magnitude spectrum
   Phase spectrum
   Illustrative problem 1.5
   Time shifted signal
   Same magnitude, Different phase
   Try it by yourself with Matlab !
Sampling Theorem
   Basis for the relation between continuous-
time signal and discrete-time signals
   A bandlimited signal can be completely
described in terms of its sample values taken
at intervals Ts as long as Ts  1/(2W)
x(t)                                   X(f)
1

-W              W   f
Ts
Impulse sampling
   Sampled waveform
                   
    x (t )  x(t )   (t  nTs )     x(nT ) (t  nT )
s            s
n                 n 

   Take FT                                                                 
X  ( f )  F [ x (t )]  F [ x(t )   (t  nTs )]  X ( f )  F [   (t  nTs )]
n                                    n 
                     
1                 n    1                      n
 X ( f )
Ts

n 
 (f  ) 
Ts   Ts

n 
X(f 
Ts
)

x(t)                                                                     X(f)
1/Ts

f
-W           W       1/Ts
Ts                                                 1/(2Ts)
Reconstruction of signal
   Low pass filter
   With Bandwidth 1/(2Ts) and Gain of Ts

X(f)         Low pass filter

f
-W     W       1/Ts
1/(2Ts)
Reconstruction from sampled signal

   If we have sampled values
   {… x(-2Ts), x(-Ts), 0, x(Ts), x(2Ts) …}
   With Nyquist interval (or Nyquist rate)
   Ts = 1/(2W)
   Then we can reconstruct x(t) using

    x(t )     x(nT )sinc(2W (t  nT ))
n 
s              s

   Example: Figure 1.17 at page 24
Aliasing or Spectral folding
   If sampling rate is Ts > 1/(2W)
   Spectrum is overlapped
   We can not reconstruct original signal with
under-sampled values
   Anti-aliasing methods are needed

X(f)                           X(f)
1/Ts

f
-W                W     f           -W      W
1/Ts
Discrete Fourier Transform
   DFT of discrete time sequence x[n]


 X d ( f )   x[n]e
 j 2 fnT
s

n 

   Relation between FT and DFT
   X ( f )  Ts X d ( f ),   for f  W

   FFT(Fast Fourier Transform)
   Efficient numerical method to compute DFT
   Example: Try Illustrative problem 1.6 by yourself
DFT in Matlab
   fft.m and ifft.m with finite samples
N
   Definition of DFT: X (n)   x[k ]e j 2 nk / N
k 1

1 N
   Definition of IDFT:       x[k ]   X (n)e j 2 nk / N
N n 1

   Time and frequency is not appeared explicitly
   Just definition implemented on a computer to compute N values for
the DFT and IDFT
   N is chosen to be N=2m
   Zero padding is used if N is not power of 2
FFT in matlab
   A sequence of length N=2m of samples of x(t)
taken at Ts
   Ts satisfies Nyquist condition
   Ts is called time resolution
   FFT gives a sequence of length N of sampled
Xd(f) in the frequency interval [0, fs=1/Ts]
   The samples are apart f  fs / N
   f is called frequency resolution
   Frequency resolution is improved by increasing N
Some remarks on short signal
   FT works on signals of infinite duration
   But, we only measure the signal for a short time

   FFT works as if the data is periodic all the time
Some remarks on short signal
   Sometimes this is correct
Some remarks on short signal
   Sometimes wrong
Frequency leakage
   If the period exactly fits the measurement time, the
frequency spectrum is correct
   If not, frequency spectrum is incorrect

Frequency domain analysis of LTI system

   The output of LTI system
   y(t )  x(t )  h(t )

   Take FT (using convolution theorem)
   Y ( f )  X ( f )H ( f )
   Where the Transfer Function of the system

   H ( f )  F[h(t )]   h(t )e j 2 ft dt


   The relation between input-output spectra
    Y( f )  X ( f ) H( f )
Y ( f )  X ( f )  H ( f )
Homeworks
   Illustrative problem 1.7
   Problems
   1.10, 1.12, 1.14, 1.15
More on Sampling
   Most real signals are analog
   The analog signal has to be converted to digital

   Information lost during this procedure (Quantization
Error)
   Inaccuracies in measurement
   Uncertainty in timing
   Limits on the duration of the measurement
More on Sampling
   Continuous analog signal has to be held
before it can be sampled
More on Sampling
   The sampling take place at equal interval of
time after the hold
   Need fast hold circuit
   Signal is not changing during the time the circuit is
acquiring the signal value
   Unless, ADC has all the time that the signal is held to
make its conversion

   We don’t know what we don’t measure
More on Sampling
   In the process of measuring signal, some
information is lost
Aliasing
   We only sample the signal at intervals
   We don’t know what happens between the
samples
Aliasing
   We must sample fast enough to see the most
rapid changes in the signal
   This is Sampling theorem

   If we do not sample fast enough
   Some higher frequencies can be incorrectly
interpreted as lower ones
Aliasing
   Called “aliasing” because one frequency
looks like another
Aliasing
   Nyquist frequency
   We must sample faster than twice the frequency
of the highest frequency component
Antialiasing
   We simply filter out all the high frequency
components before sampling
   Antialias filters must be analog
   It is too lte once you have done the sampling
More on sampling Theorem
   The sampling theorem does not say the
samples will look like the signal
More on Sampling Theorem
   Sampling theorem says there is enough
information to reconstruct the signal
   Correct reconstruction is not just draw straight
lines between samples
More on Sampling Theorem
   The impulse response of the reconstruction
filter has sinc (sinx/x) shape
   The input to the filter is the series of discrete
impulses which are samples
   Every time an impulse hits the filter, we get
‘ringing’
   Superposition of all these rings reconstruct the
proper signal
Frequency resolution
   We only sample the signal for a certain time
   We must sample for at least one complete cycle
of the lowest frequency we want to resolve
Quantization
   When the signal is converted to digital form
   Precision is limited by the number of bits
available
Uncertainty in the clock
   Uncertainty in the clock timing leads to
errors in the sampled signal
Digitization errors
   The errors introduced by digitization are both
nonlinear and signal dependent
   Nonlinear
   We can not calculate their effect using normal maths.
   Signal dependent
   The errors are coherent and so cannot be reduced by
simple means
Digitization errors
   The effect of quantiztion error is often similar
to an injection of random noise

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