1. Electrical Communications Systems ECE.09.433
Signals and Spectra II
Dr. Shreek Mandayam
Electrical & Computer Engineering
Rowan University

2. Plan CFT’s (spectra) of common waveforms Discrete Fourier Transform
Impulse
Sinusoid
Rectangular Pulse
Discrete Fourier Transform
How to get the frequency axis in the DFT

3. ECOMMS: Topics

4. Continuous Fourier Transform
Continuous Fourier Transform (CFT)
Frequency, [Hz]
Amplitude
Spectrum
Phase
Inverse Fourier Transform (IFT)
See p. 46
Dirichlet Conditions

5. CFT’s of Common Waveforms
Impulse (Dirac Delta)
Sinusoid
Rectangular Pulse
Matlab Demo:
recpulse.m

6. CFT for Periodic Signals
Recall:
FS: Periodic Signals
CFT: Aperiodic Signals
We want to get the CFT for a periodic signal
What is ?

7. Discrete Fourier Transform (DFT)
Equal time intervals
Discrete Domains
Discrete Time: k = 0, 1, 2, 3, …………, N-1
Discrete Frequency: n = 0, 1, 2, 3, …………, N-1
Discrete Fourier Transform
Inverse DFT
Equal frequency intervals
n = 0, 1, 2,….., N-1
k = 0, 1, 2,….., N-1

8. Importance of the DFT
Allows time domain / spectral domain transformations using discrete arithmetic operations
Computational Complexity
Raw DFT: N2 complex operations (= 2N2 real operations)
Fast Fourier Transform (FFT): N log2 N real operations
Fast Fourier Transform (FFT)
Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the periodicity and symmetry of e-j2pkn/N
VLSI implementations: FFT chips
Modern DSP

9. How to get the frequency axis in the DFT
The DFT operation just converts one set of number, x[k] into another set of numbers X[n] - there is no explicit definition of time or frequency
How can we relate the DFT to the CFT and obtain spectral amplitudes for discrete frequencies?
(N-point FFT)
Need to
know fs
n= n=N
f= f = fs

10. All those signals………. Amplitude w(t) time Time Amplitude Signal 111
continuous continuous continuous-time
analog signal
discrete discrete discrete-time
digital signal Cn
111
110
101
100
011
010
001
000
sampling
Amplitude
time
discrete continuous discrete-time
analog signal
w(nTs) Ts
sampling
discrete continuous discrete-time
sequence
w[n] n=
indexing

11. …..and all those transforms
Sample in time,
period = Ts
Continuous-time
analog signal
w(t)
Discrete-time
analog sequence
w [n] C D
Continuous-variable
Discrete-variable
Laplace
Transform
W(s)
Continuous
Fourier
Transform
W(f)
z-Transform
W(z)
Discrete-Time
Fourier
Transform
W(W)
Discrete
Fourier
Transform
W(k)
z = ejW
s = jw
w=2pf
=2pf
W = w Ts,
scale
amplitude
by 1/Ts
Sample in
frequency,
W = 2pn/N,
N = Length
of sequence

12. Summary