ECEN3513 Signal Analysis Lecture #9 11 September 2006 n Read section 2.7, 3.1, 3.2 (to top of page 6) n Problems: 2.7-3, 2.7-5, 3.1-1

ECEN3513 Signal Analysis Lecture #10 13 September 2006 n Read section 3.4 n Problems: 3.1-5, 3.2-1, 3.3a & c n Quiz 2 results: Hi = 10, Low = 5.5, Ave = 7.88 Standard Deviation = 1.74

MathCad Correlation Solution

MathCad Convolution Solution

MathCad Correlation Solution

You can't always trust your software tools!

Generating a Square Wave... 5 cycle per second square wave

Generating a Square Wave vp 5 Hz 1/3 vp 15 Hz

Generating a Square Wave /5 vp 25 Hz Hz + 15 Hz

Generating a Square Wave /7 vp 35 Hz Hz + 15 Hz + 25 Hz

Generating a Square Wave Hz + 15 Hz + 25 Hz + 35 Hz cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t) 5 cycle per second square wave generated using 4 sinusoids

Generating a Square Wave... 5 cycle per second square wave generated using 50 sinusoids

Generating a Square Wave... 5 cycle per second square wave generated using 100 sinusoids

Fourier Series (Trigonometric Form) n x(t) = a 0 + ∑ (a n cos nω 0 t + b n sin nω 0 t) n a 0 = (1/T) x(t)1 dt n a n = (2/T) x(t)cos nω 0 t dt n b n = (2/T) x(t)sin nω 0 t dt n=1 ∞ T T T x(t) periodic T = period ω 0 = 2π/T

Fourier Series (Harmonic Form) n x(t) = a 0 + ∑ c n cos(nω 0 t - θ n ) n a 0 = (1/T) x(t)1 dt n c n 2 = a n 2 + b n 2 n θ n = tan -1 (b n /a n ) n=1 ∞ T x(t) periodic T = period ω 0 = 2π/T

Fourier Series (Exponential Form) n x(t) = ∑ d n e jnω 0 t n d n = (1/T) x(t)e -jnω 0 t dt n= -∞ ∞ T x(t) periodic T = period ω 0 = 2π/T

Transforms X(s) = x(t) e -st dt 0-0- ∞ Laplace ∞ X(3) = x(t) e -3t dt 0-0-

Transforms X(f) = x(t) e -j2πft dt -∞-∞ ∞ Fourier n e -j2πft = cos(2πft) - j sin(2πft) n Re[X(f)] = similarity between cos(2πft) & x(t) u Re[X(10.32)] = amount of Hz cosine in x(t) n Im[X(f)] = similarity between sin(2πft) & x(t) u -Im[X(10.32)] = amount of Hz sine in x(t)

Fourier Series n Time Domain signal must be periodic n Line Spectra u Energy only at discrete frequencies Fundamental (1/T Hz) Harmonics (n+1)/T Hz; n = 1, 2, 3,... n Spectral envelope is based on FT of periodic base function

Fourier Transforms X(f) = x(t) e -j2πft dt -∞-∞ ∞ Forward x(t) = X(f) e j2πft df -∞-∞ ∞ Inverse

Fourier Transforms X(ω) = x(t) e -jωt dt -∞-∞ ∞ Forward x(t) = X(ω) e jωt dω 2π -∞-∞ ∞ Inverse

Fourier Transforms Fourier Transforms n Basic Theory n How to evaluate simple integral transforms n How to use tables n On the job (BS signal processing or Commo) u Mostly you’ll use Fast Fourier Transform Info above will help you spot errors u Only occasionally will you find FT by hand Masters or PhD may do so more often

Summing Complex Exponentials (T = 0.2 seconds) f (Hz) ∑e -jn2πf/T n =

Summing Complex Exponentials (T = 0.2 seconds) f (Hz) ∑e -jn2πf/T n = -1 n =

Summing Complex Exponentials (T = 0.2 seconds) f (Hz) ∑e -jn2πf/T n = -5 n =

Summing Complex Exponentials (T = 0.2 seconds) f (Hz) ∑e -jn2πf/T n = -10 n =

Summing Complex Exponentials (T = 0.2 seconds) f (Hz) ∑e -jn2πf/T n = -100 n =

Summing Complex Exponentials (T = 0.2 seconds) f (Hz) ∑e -jn2πf/T n = -100 n =