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

2. Plan Digital and Analog Communications Systems
Properties of Signals and Noise
Terminology
Power and Energy Signals
Continuous Fourier Transform (CFT)

3. ECOMMS: Topics

4. Communications Systems
Digital
Finite set of messages (signals)
inexpensive/expensive
privacy & security
data fusion
error detection and correction
More bandwidth
More overhead (hw/sw)
Analog
Continuous set of messages (signals)
Legacy
Predominant
Inexpensive

5. Signal Properties: Terminology
Waveform
Time-average operator
Periodicity
DC value
Power
RMS Value
Normalized Power
Normalized Energy

6. Power and Energy Signals
Power Signal
Infinite duration
Normalized power is finite and non-zero
Normalized energy averaged over infinite time is infinite
Mathematically tractable
Energy Signal
Finite duration
Normalized energy is finite and non-zero
Normalized power averaged over infinite time is zero
Physically realizable
Although “real” signals are energy signals, we analyze them pretending they are power signals!

7. The Decibel (dB) Measure of power transfer
1 dB = 10 log10 (Pout / Pin)
1 dBm = 10 log10 (P / 10-3) where P is in Watts
1 dBmV = 20 log10 (V / 10-3) where V is in Volts

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

9. Properties of FT’s If w(t) is real, then W(f) = W*(f)
If W(f) is real, then w(t) is even
If W(f) is imaginary, then w(t) is odd
Linearity
Time delay
Scaling
Duality
See p. 52
FT Theorems

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

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

12. 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

13. 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

14. 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

15. Summary