CoverageChapter 2: Important ConceptsChapter 3: Important ConceptsChapter 4: Important ConceptsChapter 5: Important ConceptsChapter 6: Important ConceptsChapter 7: Important ConceptsChapter 8: Important ConceptsChapter 9: Not on ExamChapter 10: Not on ExamCoverage 1 ECE 2610 Final ReviewDr. Wickert, Spring 2011CoverageChapter 2: Important Concepts• Basic sine and cosine function properties, e.g., and other• A continuous-time sinusoidal signal, , is controlled by three parameters– The period s and frequency Hz and rad/s both are inversely related to theperiod• Time shifting waveforms to the left or right by , e.g., – For a sinusoid time shift is related to phase shift (notes p. 2-12)• Complex number arithmetic, must know this!– Rectangular form ( ) for adding and subtracting– Polar form multiplying and dividing– Euler’s formula and Euler’s inverse formulas (very useful in ECE)• Complex exponential signals, – Rotating phasor interpretation– Phasor addition rule in the sum of sinusoids:Chapter Text Sections Notes Pages2 2-1—2-6, Appendix A 2-1—2-33, Quiz 1 & 23 3-1—3-8 3-1—3-37, 3-41—3-45, Quiz 3 & 44 4-1—4-2, 4-4—4-5 4-1—4-21, Quiz 55 5-1—5-7 5-1—5-38, Quiz 6 & 76 6–1—6–8 6–1—6–32, Quiz 8 & 97 7–1—7–9 7–1—7–28, Quiz 108 8–1—8–11 8–1—8–47, Quiz 11a, 11b, 12a, 12bsin 2–cos=A 0t +cosT0f00t0yt xt t0–=xjy+rejAej 0t +Ak0t k+cosk 1=NA 0t +cos=ECE 2610 Final ReviewChapter 3: Important Concepts 2 where we find and usingChapter 3: Important Concepts• Two-sided line spectrum for a sum of sinusoids obtained from a sequence of frequency/com-plex amplitude pairs that correspond to the of each sinusoid and a complexamplitude that corresponds to the magnitude and phase of each sinusoid– Euler’s formula key to making this connection– DC/constant/zero frequency terms are treated as a special case• Beat notes and AM, and how the sum of two sinusoidal signals is related to the product oftwo sinusoidal signals• Periodic waveforms and Fourier series– Fourier synthesis formula (sum complex exponentials to approximate )– Fourier analysis formula (integrate to get coefficients)– Simple Fourier series properties, such as computing – The spectrum of a Fourier series (harmonically related sinusoids having a fundamentalfrequency)– Differences in convergence properties for say a square wave versus a triangle wave• FM chirp signals (linear chirp)– Instantaneous frequency– The spectrogramChapter 4: Important Concepts• Sampling sinusoidal signals to form sinusoidal sequences– via – The choice of sampling rate and the sampling theorem– Alias frequencies and the principal alias band or – The concept of the folding frequency and viewing the alias frequencies as in the Figureon notes p. 4-13– Aliasing in a linear chirp signal• Ideal reconstruction and the ideal C-to-D converter, we map from back to frequency inHz using AAkejkk 1=NAej=frequencyxta002f0= 0ˆTs2f0fs== – ffs– 2 fs2f fs2=ECE 2610 Final ReviewChapter 5: Important Concepts 3 – zero-order hold versus linear interpolation– The spectrum view of sampling and reconstruction, i.e., the D-to-C keeps only those fre-quencies on the principle alias interval for reconstructionChapter 5: Important Concepts• Finite impulse response filtering using a simple moving average filter and the feed-forwarddifference equation– Calculating the output using a simple table (notes p. 5-3)• The general FIR filter• Unit impulse sequence, , where is an arbitrary sequence (time) shift• The impulse response, • The delay system, • Convolution sum view of FIR filtering– Calculating the output of an FIR filter using the convolution sum view, and a (more thanone works) table approach• The of MATLAB’s filter function to numerically obtain the output of an FIR filter given and the filter coefficients ; be familiar with how this works so you can interpret simplecode statements• Implementation of FIR filters using the building blocks of a multiplier, adder, and unit delay– Difference equation to block diagrams (direct form) and back– Other forms converting to difference equation• Linear time invariant systems– Proving whether or not a system in time invariant– Proving whether or not a system is linear• Convolution and LTI systems– Convolution operator– Convolution is commutative– Convolution is associative• Cascaded LTI systems: • Moving average filters as an exampleMIDTERM STOPPING POINT nn0–n0hnyn xn n0–=xnbkhn h1n*h2n=ECE 2610 Final ReviewChapter 6: Important Concepts 4 Chapter 6: Important Concepts• Definition of frequency response for an FIR filter • Fundamental result for real sinusoids is given input the output is• Transient response is visible when the sinusoidal input is applied starting at asopposed to for the steady-state case; For an FIR filter the transient only lasts for Msamples • Frequency response properties– Close relationship between the difference equation, impulse response, and the frequencyresponse– Periodicity of – Conjugate symmetry• Cascaded systems: • The moving (running) average filter which has periodic nulls at multiples of ,where is the total number of filter taps• Filtering continuous-time signals by forming an C-to-D– –D-to-C system– Need to be aware of how sampling theory applies in such a system– Fundamental concept is used to convert between the two frequencyaxes– When input passes through a DSP, provided it is not aliased, itis returned asChapter 7: Important Concepts• The definition for finite duration sequences and FIR filtersfor the support interval of being • For FIR filters having impulse response , the z-transform of produces the systemHejˆ bkejˆk–k 0=Mhkejˆk–k 0=M==xn A ˆ0n +cos=yn AH ejˆ0 ˆ0n  Hejˆ0++cos=n 0=n –=HejˆHejˆ H1ejˆH2ejˆ=2 M 1+M 1+HejˆˆTs2ffs==xt A 2f0t +cos=yt AH ej2f0fs2f0 Hej2f0fs++cos=Xz